mirror of https://gitee.com/XmacsLabs/planet.git
1101 lines
361 KiB
Plaintext
1101 lines
361 KiB
Plaintext
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<TeXmacs|2.1.3>
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<style|<tuple|projector|reddish|number-long-article|narrow-multiply|python|hanging-theorems>>
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<\body>
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<\hide-preamble>
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<assign|math|<macro|x|<math-colored|<with|math-display|true|<uncolored-math|<with|font-base-size|12|<arg|x>>>>>>>
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<assign|equation*|<\macro|x>
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<math-colored|<\uncolored-equation*>
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<inactive|<with|font-base-size|11|<arg|x>>>
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</uncolored-equation*>>
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</macro>>
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<assign|equations-base|<\macro|x>
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<math-colored|<\uncolored-equations-base>
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<inactive|<with|font-base-size|11|<arg|x>>>
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</uncolored-equations-base>>
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</macro>>
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<assign|infix-or|<macro|<math-or|<text|
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<inactive|<with|font-base-size|10|or>>>>>>
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<assign|text|<macro|body|<with|mode|text|<with|font-base-size|10|<arg|body>>>>>
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<assign|nbsp|<macro| <no-break><specific|screen|<resize|<move|<with|color|#A0A0FF|
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>|-0.3em|>|0em||0em|>>>>
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</hide-preamble>
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<\slideshow>
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<\slide>
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<doc-data|<doc-title|Chapter 3: Differentiation Rules>>
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<\table-of-contents|toc>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|1<space|2spc>Derivatives
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of Polynomials and Exponential Functions>
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-1><vspace|0.5fn>
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<with|par-left|1tab|1.1<space|2spc>Power functions
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-2>>
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<with|par-left|1tab|1.2<space|2spc>Linear combination
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-3>>
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<with|par-left|1tab|1.3<space|2spc>Exponential functions
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-4>>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|2<space|2spc>The
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Product and Quotient Rules> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-5><vspace|0.5fn>
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<with|par-left|1tab|2.1<space|2spc>The product rule
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-6>>
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<with|par-left|1tab|2.2<space|2spc>The quotient rule
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-7>>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|3<space|2spc>Derivatives
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of Trigonometric Functions> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-8><vspace|0.5fn>
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<with|par-left|1tab|Derivative of
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<with|color|black|font-family|rm|<with|math-display|true|<with|mode|math|<with|font-base-size|12|sin
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x>>>> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-9>>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|4<space|2spc>The
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Chain Rule> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-10><vspace|0.5fn>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|5<space|2spc>Implicit
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differentiation> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-11><vspace|0.5fn>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|6<space|2spc>Derivatives
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of Logarithmic Functions> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-12><vspace|0.5fn>
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<with|par-left|1tab|6.1<space|2spc>Logarithmic differentiation
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-13>>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|7<space|2spc>Rates
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of Change in the Economics and Social Sciences>
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-14><vspace|0.5fn>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|8<space|2spc>skip>
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-15><vspace|0.5fn>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|9<space|2spc>skip>
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-16><vspace|0.5fn>
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<vspace*|1fn><with|font-series|bold|math-font-series|bold|10<space|2spc>Linear
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Approximations and Differentials>
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-17><vspace|0.5fn>
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<with|par-left|1tab|10.1<space|2spc>Linear approximations
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-18>>
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<with|par-left|1tab|10.2<space|2spc>Differentials
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<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
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<no-break><pageref|auto-19>>
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</table-of-contents>
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<section|Derivatives of Polynomials and Exponential Functions>
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<subsection|Power functions>
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<\itemize>
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<item>If <math|f<around*|(|x|)>=c>, where <math|c> is a constant.
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Then
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<\equation*>
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f<rprime|'><around*|(|x|)>=lim<rsub|h\<rightarrow\>0>
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<dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
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<dfrac|c-c|h>=0.
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</equation*>
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<item>If <math|f<around*|(|x|)>=x>, then
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<\equation*>
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f<rprime|'><around*|(|x|)>=lim<rsub|h\<rightarrow\>0>
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<dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
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<dfrac|<around*|(|x+h|)>-x|h>=1.
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</equation*>
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<item>If <math|f<around*|(|x|)>=x<rsup|2>>, then
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<\equation*>
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f<rprime|'><around*|(|x|)>=lim<rsub|h\<rightarrow\>0>
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<dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
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<dfrac|<around*|(|x+h|)><rsup|2>-x<rsup|2>|h>
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</equation*>
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<\equation*>
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=lim<rsub|h\<rightarrow\>0> <dfrac|2*x*h+h<rsup|2>|h>=lim<rsub|h\<rightarrow\>0>
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<around*|(|2x+h|)>=2*x.
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</equation*>
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<item>In general, if <math|f<around*|(|x|)>=x<rsup|n>,n\<in\>\<bbb-N\>>,
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then
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<\equation*>
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f<rprime|'><around*|(|x|)>=n*x<rsup|n-1>.
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</equation*>
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Hint: use the binomial expansion
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<\equation*>
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<around*|(|a+b|)><rsup|n>=a<rsup|n>+n*a<rsup|n-1>*b+<dfrac|n<around*|(|n-1|)>|2>*a<rsup|n-2>*b<rsup|2>+\<cdots\>+b<rsup|n>=<big|sum><rsub|k=0><rsup|n><matrix|<tformat|<table|<row|<cell|n>>|<row|<cell|k>>>>>*a<rsup|k>*b<rsup|n-k>,
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</equation*>
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where
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<\equation*>
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<matrix|<tformat|<table|<row|<cell|n>>|<row|<cell|k>>>>>=<dfrac|n!|k!*<around*|(|n-k|)>!>.
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</equation*>
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<item>The above results can be generalized to
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<math|f<around*|(|x|)>=x<rsup|a>,a\<in\>\<bbb-R\>>.\
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<\equation*>
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<marked|f<rprime|'><around*|(|x|)>=a*x<rsup|a-1>>.
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</equation*>
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</itemize>
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\;
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From this result we can find the derivative of any polynomial.
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<subsection|Linear combination>
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<\equation*>
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<marked|<around*|[|c<rsub|1>*f<around*|(|x|)>+c<rsub|2>*g<around*|(|x|)>|]><rprime|'>=c<rsub|1>*f<rprime|'><around*|(|x|)>+c<rsub|2>*g<rprime|'><around*|(|x|)>>.
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</equation*>
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<\example>
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If <math|f<around*|(|x|)>=x<rsup|100>+5*x<rsup|3>+6*x-12,>then
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<\equation*>
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f<rprime|'><around*|(|x|)>=100*x<rsup|99>+15*x<rsup|2>+6
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</equation*>
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</example>
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<subsection|Exponential functions>
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Let <math|f<around*|(|x|)>=b<rsup|x>,b\<gtr\>0>. Then
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<\eqnarray*>
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<tformat|<table|<row|<cell|f<rprime|'><around*|(|x|)>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
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<dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
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<dfrac|b<rsup|x+h>-b<rsup|x>|h>=lim<rsub|h\<rightarrow\>0>
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<dfrac|b<rsup|x>*<around*|(|b<rsup|h>-1|)>|h>>>|<row|<cell|>|<cell|=>|<cell|b<rsup|x>*lim<rsub|h\<rightarrow\>0>
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<dfrac|*b<rsup|h>-1|h>.>>>>
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</eqnarray*>
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In fact,\
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<\equation*>
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lim<rsub|h\<rightarrow\>0> <dfrac|*b<rsup|h>-1|h>=f<rprime|'><around*|(|0|)>
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</equation*>
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if it exists, and
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<\equation*>
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f<rprime|'><around*|(|x|)>=f<rprime|'><around*|(|0|)>*b<rsup|x>.
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</equation*>
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In fact,\
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<\equation*>
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<marked|<around*|(|b<rsup|x>|)><rprime|'>=<around*|(|ln
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b|)>*b<rsup|x>>.
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</equation*>
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In particular,
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<\equation*>
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<marked|<around*|(|e<rsup|x>|)><rprime|'>=e<rsup|x>>.
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</equation*>
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<\question>
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Prove the limit exists. Hint: tranform the limit to the form
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containing the term <math|lim<rsub|x\<rightarrow\>\<infty\>><around*|(|1+<dfrac|1|x>|)><rsup|x>>.
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</question>
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<hlink|Click Here|https://planetmath.org/convergenceofthesequence11nn>
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to see a proof that the limit of the sequence
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<math|<around*|(|1+<dfrac|1|n>|)><rsup|n>>exists as
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<math|n\<rightarrow\>\<infty\>>.
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\;
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<\definition>
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The <strong|relative rate of change> of a function
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<math|f<around*|(|x|)>> is
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<\equation*>
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<dfrac|f<rprime|'><around*|(|x|)>|f<around*|(|x|)>>,
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</equation*>
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and the <strong|percentage rate of change> is
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<\equation*>
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100*<dfrac|f<rprime|'><around*|(|x|)>|f<around*|(|x|)>>.
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</equation*>
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</definition>
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\;
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<section|The Product and Quotient Rules>
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<subsection|The product rule>
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\;
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Motivation: How to find the derivative function of
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<math|f<around*|(|x|)>=x<rsup|2>*e<rsup|x>>?
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\;
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<\theorem>
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<dueto|Product Rule>If <math|f> and <math|g> are both differentiable
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at <math|x>, then <math|f*g> is differentiable at <math|x>. Moreover,
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<\equation*>
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<around*|(|f*g|)><rprime|'><around*|(|x|)>=f<rprime|'><around*|(|x|)>*g<around*|(|x|)>+f<around*|(|x|)>*g<rprime|'><around*|(|x|)>.
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</equation*>
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</theorem>
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<\proof>
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From the definition,
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<\eqnarray*>
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<tformat|<table|<row|<cell|<around*|(|f*g|)><rprime|'><around*|(|x|)>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
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<dfrac|f<around*|(|x+h|)>*g<around*|(|x+h|)>-f<around*|(|x|)>*g<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|f<around*|(|x+h|)>*g<around*|(|x+h|)><with|color|blue|-f<around*|(|x|)>*g<around*|(|x+h|)>+f<around*|(|x|)>*g<around*|(|x+h|)>>-f<around*|(|x|)>*g<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|g<around*|(|x+h|)>*<around*|[|f<around*|(|x+h|)>-f<around*|(|x|)>|]>+f<around*|(|x|)>*<around*|[|g<around*|(|x+h|)>-g<around*|(|x|)>|]>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|g<around*|(|x+h|)>*<around*|[|f<around*|(|x+h|)>-f<around*|(|x|)>|]>|h>+lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|f<around*|(|x|)>*<around*|[|g<around*|(|x+h|)>-g<around*|(|x|)>|]>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
g<around*|(|x+h|)>*lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>+lim<rsub|h\<rightarrow\>0>
|
||
|
|
f<around*|(|x|)>*lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|g<around*|(|x+h|)>-g<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|g<around*|(|x|)>*f<rprime|'><around*|(|x|)>+f<around*|(|x|)>*g<rprime|'><around*|(|x|)>.>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
\;
|
||
|
|
</proof>
|
||
|
|
|
||
|
|
Intuitive explanation
|
||
|
|
|
||
|
|
<\wide-block>
|
||
|
|
<tformat|<cwith|1|1|2|2|cell-valign|c>|<cwith|1|1|1|1|cell-valign|c>|<table|<row|<\cell>
|
||
|
|
<image|<tuple|<#89504E470D0A1A0A0000000D49484452000001E00000011E0806000000524CC6FC000000097048597300000B1300000B1301009A9C180000200049444154789CECBD69945CD77DE0F7BBCB7BAF965E806EA0BBB1121B0110E0029058484AB445C996154B91272325236732B6E78CED4C3ECD71F2C11F7CFCDD39394E664E72E29978896734963D5EA4634BD1C8634926417001091200897DDF1B6B03E8A5AADE72EFCD87FB5E5575A3B1123401E1FD701A555DF5F67EEFFEEF7F87929292929292927F708473CE7DDA07712B9C73A4698A738E288A30C660AD456B8D100E6C066480F03F227F85AED759B7FC491F7A494949C903C441F7502DBADFCC18EB9C68AF821320C0298BC360F3B14F2010C8691B13D3B6337D8C148FED902900852BAE95EB7C5A60AD052110129CCBC8D204A51452FABF8D901ADAD77A3AFA133CF28F8D1002ADB53FC11CA51400CEFAEF4131ED2614028768DF846256396C67FBB0A4A4A4E4E1E59652B010B833940FD1F51E017E64ECFABD5BF08A5904F06C02FF71A3B8365DBFC2B44B23B52469C5202CD659B492F96A022124CEB95BC8A1875C0306AF0577BF0A21C8B20C6B2D815608E9CFAC98D781E89AE3DD82C7773A575252F288226E39541763602E50BBE5A62BE4B2C3094B4772080AADACBDBC98298067ECE1711D3645F7A9774B603FA971D6618D4169496652ACC90087521AA9140285906AD64D3FD41AB0B516E71C4A299C734829C9B20CE71C41102084C0E682D6F96B014CB7C0CCCE633B9D2B29297904F1CAECEDC72DD735FE09BA85AFDF82607621306D1FB7F8EDB115BE00B85C08BBB67AD7FE1C88E31661A43136236EC5682DD15AFB751C4839BBF9191E720DD839374D03564A91651952CADB9E544949494949C9A741AB39899402AD0384B8B5F60B0FB9062CF2199F31A6FD99520A210467CE9CE5F8C953C4AD164276CCCDA2CB185D70F39CA563462875E1929292478D9BBD6802D1652A9D39B289AEFFDD4CFFEE6CEAADEB2C3FCB178F0F022C16D7B1D37B19E33A0B2809719CA09522AA54D9B8E1597A7BFB31993745DF4EC63CD402B8D080A5F48E6C630C4A299ACD267FFA677FC6DFFCCD77191FBF8190B2B0C603DDDEDF99378B98E59392929292879D5946AE2E77EECD9E493F0E5AE790422085F02EBBDCAD57ACE3C0DB49BBA5774917AE4BF8CE8E00B22C4369451068FEE7DFF857FCC257FF1BAAD5FAB400E2D978A805708194B273E300376EDCE0BFFCED0F786BFB36AA518496126B3294944829B026C33A8B6CA72675EED49BF5E39B6FDEEEF7777ABD15B75BFE76EBCEB61EB37C76BBF56FB5FCFD1EF39DD6BF9BE53FEE39DFCD7AB73B965B712FE77A2FF7C7CCE5EF657F1FF79C6FB7DEC37ACE7773AEF77BCEF7F30CDDCFB5BA9BE5EFFF7E9CAE4989692797EBB7D28F6FC658A452186B892A21AD2425CD0C99B154AA558484248901471868849224498298E6D6BBF98A3F8ECA4A612B2802D960BA1A279C5714C320246E25A489E3073FF8019BB7BCC4AA954FDEF19A3DD4025874051D887C06073E38ABD56CD21305BCF2E266D6AD5943A82581922801691AE39C454B8170968EEB5CE2F234252740CEB4C4DCEA38B8F9619EF9DDDDAECFDDEDF2A66DDFCD7E6EB5CFBB3DBEBB5DFF76DBBC97FDDEEDB1DCED7ADCC3BE6F3730DFCFBEEF67FDDB4D3AEE767DEE71F98F7BCE9FC4B5FA249FA1D9B671BFEBDDCD7D7FB7EBDDEB39CF0C0092AEB394E88A40B552D08A53C27A95244DE9EDEB63EFC1837CF0D13E265A312F7FF6332C5BF10471DCC4D8844A25042998989C20AA44DEEA28249D2B2D018770330390EE78C0777FE2B75BEFDE2FD4DD73170F8413608497174EE4D7C179C12100E10481D2B80CFEEE6F7FC4D933A334E3982CCD009077089C7BA8057041770A1280D29A9E5A1593A6AC5FBD825FF9E5FF9EE1F9F37049134C8A3506211D4A82CB0C028113D25F3221DB331AE1C40C5FCA9DE6A78F020FEB393CACC7752FDCEB3994E7FC68F2B09D83C0D72EE8983385C38F6B88B69AE60420BD361BD6ABA440B556E74FBEF527ECFA70378E849FFADC4B7CE517BE827319CDD6145283C3D26C35A8D4AA58E7707E43F98F9748A2AD0B765108FEBB51F9675ED27B597EB63FC7EDD663966DDC6EBD99CBE7AFC28115022B240E997FE6F2ABEEC0390492DE7A2F8DF106EFBCF51E8DD609EAF53A611802608D03495E94E3661E09017C13CEE1ACC158E8A95458303C8FB90373B19312890167218BF3BBB430ABE4053B8ADF0B9F879BEDAFD0FDFE4EAFF7BADE6CCBDC693DEE719B775AFE419DF3DDEE67B665EFF79C6FB7DEDD9CC3A771CEF7B39F0771CEF7BAAD87E19CEF66BDFBB91E773A8799EB3D4CE72C4018FFBBEB5ADEE53F004280D6FE354BB1D221C30A0C0E126A885B2D621353EFAF33387F2E49DA44371C6887508E1E222FDE45B14D81CDC7CCE2B75BE721FF24E2AFBD436191DE2AE0FC2742382F5A9C0324B5A8C2BCB903582750D25FBF2C33EDADF8BF9998752F8FA600CEA9281026A371E52AB23989499AD42B2181B0A4CD29A4001DE83C98CB5F4427140E7078ED57BA5B5D9A929292924F1FEF32B3E40E473F66B91995AB72D9ECA4C44AC144D2C409C9602DC29A041D58B22CA31137B97A638C666B82346B21942533092A10186B71428010382773012C913884B3DCD104FD13884362D12024C2FA4F44618A06A4135CCB1C4B163CE17DEF52E701C3DE5AA1D4EDD3651F69019C1A68365B586BA946158CB0480CC259C228F4F157D6E67958B2EDDBB0F9ABB0B7AAD059525252F270E0C59E002C48AF7DB5B5DF228239AFC664B084B55E7A02C554AB012E43389F1F92598B4580140
|
||
|
|
</cell>|<\cell>
|
||
|
|
Let <math|u=f<around*|(|x|)>,v=g<around*|(|x|)>>. For small change
|
||
|
|
<math|\<Delta\>x>, let <math|\<Delta\>u,\<Delta\>v> be the change
|
||
|
|
in <math|u,v>, respectively. Let <math|\<Delta\><around*|(|u*v|)>>
|
||
|
|
be the change in <math|u*v>.
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|\<Delta\><around*|(|u*v|)>>|<cell|=>|<cell|<around*|(|u+\<Delta\>u|)>*<around*|(|v+\<Delta\>v|)>-u*v>>|<row|<cell|>|<cell|=>|<cell|u*v+u*\<Delta\>v+v*\<Delta\>u+\<Delta\>u*\<Delta\>v-u*v>>|<row|<cell|>|<cell|=>|<cell|u*\<Delta\>v+v*\<Delta\>u+\<Delta\>u*\<Delta\>v>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|<dfrac|\<Delta\><around*|(|u*v|)>|\<Delta\>x>>|<cell|=>|<cell|<dfrac|u*\<Delta\>v+v*\<Delta\>u+\<Delta\>u*\<Delta\>v|\<Delta\>x>>>|<row|<cell|>|<cell|=>|<cell|u*<dfrac|\<Delta\>v|\<Delta\>x>+v*<dfrac|\<Delta\>u|\<Delta\>x>+\<Delta\>u*<dfrac|\<Delta\>v|\<Delta\>x>>>|<row|<cell|>|<cell|\<rightarrow\>>|<cell|u*v<rprime|'>+v*u<rprime|'>+0<space|1em>as<space|1em>\<Delta\>x\<rightarrow\>0>>|<row|<cell|>|<cell|=>|<cell|u*v<rprime|'>+v*u<rprime|'>>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
</cell>>>>
|
||
|
|
</wide-block>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\enumerate-alpha>
|
||
|
|
<item>If <math|f<around|(|x|)>=x*e<rsup|x>>, find
|
||
|
|
<math|f<rprime|'><around|(|x|)>>.\
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<around*|(|x*e<rsup|x>|)><rprime|'>=x<rprime|'>*e<rsup|x>+x*<around*|(|e<rsup|x>|)><rprime|'>=e<rsup|x>+x*e<rsup|x>=e<rsup|x>*<around*|(|1+x|)>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<item>Find the <math|n>th derivative,
|
||
|
|
<math|f<rsup|<around|(|n|)>><around|(|x|)>>.
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|<around*|(|x*e<rsup|x>|)><rprime|''>>|<cell|=>|<cell|<around*|[|e<rsup|x>*<around*|(|1+x|)>|]><rprime|'>>>|<row|<cell|>|<cell|=>|<cell|e<rsup|x><around*|(|1+x|)>+e<rsup|x>=e<rsup|x>*<around*|(|2+x|)>.>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
Repeat this process one more time, we get\
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<around*|(|x*e<rsup|x>|)><rprime|'''>=e<rsup|x>*<around*|(|3+x|)>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
In general
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
f<rsup|<around*|(|n|)>>=e<rsup|x>*<around*|(|n+x|)>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\exercise>
|
||
|
|
Prove the general result using mathematical induction.
|
||
|
|
</exercise>
|
||
|
|
</enumerate-alpha>
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<subsection|The quotient rule>
|
||
|
|
|
||
|
|
<\theorem>
|
||
|
|
<dueto|Quotient Rule>If <math|f> and <math|g> are both differentiable
|
||
|
|
at <math|x> and <math|g<around*|(|x|)>\<neq\>0>, then <math|f*g> is
|
||
|
|
differentiable at <math|x>. Moreover,
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<around*|(|<dfrac|f|g>|)><rprime|'><around*|(|x|)>=<dfrac|f<rprime|'>*<around*|(|x|)>g<around*|(|x|)>-f*<around*|(|x|)>g<rprime|'><around*|(|x|)>|g<rsup|2><around*|(|x|)>>.
|
||
|
|
</equation*>
|
||
|
|
</theorem>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\exercise>
|
||
|
|
Prove the quotient rule using the definition of derivative.
|
||
|
|
</exercise>
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
Let <math|y=<frac|x<rsup|2>+x-2|x<rsup|3>+6>>. Then
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|y<rprime|'>>|<cell|=>|<cell|<frac|<around*|(|x<rsup|2>+x-2|)><rprime|'>*<around*|(|x<rsup|3>+6|)>-<around*|(|x<rsup|2>+x-2|)>*<around*|(|x<rsup|3>+6|)><rprime|'>|<around*|(|x<rsup|3>+6|)><rsup|2>>>>|<row|<cell|>|<cell|=>|<cell|<frac|<around*|(|2*x+1|)>*<around*|(|x<rsup|3>+6|)>-<around*|(|x<rsup|2>+x-2|)>*<around*|(|3*x<rsup|2>|)>|<around*|(|x<rsup|3>+6|)><rsup|2>>>>|<row|<cell|>|<cell|=>|<cell|<frac|2*x<rsup|4>+x<rsup|3>+12*x+6-<around*|(|3*x<rsup|4>+3*x<rsup|3>-6*x<rsup|2>|)>|<around*|(|x<rsup|3>+6|)><rsup|2>>>>|<row|<cell|>|<cell|=>|<cell|<frac|-x<rsup|4>-2*x<rsup|3>+6*x<rsup|2>+12*x+6|<around*|(|x<rsup|3>+6|)><rsup|2>>.>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
</example>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<section|Derivatives of Trigonometric Functions>
|
||
|
|
|
||
|
|
<subsection*|Derivative of <math|sin x>>
|
||
|
|
|
||
|
|
By the definition
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|<around*|(|sin
|
||
|
|
x|)><rprime|'>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|sin<around*|(|x+h|)>-sin<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|sin<around*|(|x|)>*cos<around*|(|h|)>+cos<around*|(|x|)>*sin<around*|(|h|)>-sin<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|sin<around*|(|x|)>*<around*|[|cos<around*|(|h|)>-1|]>+cos<around*|(|x|)>*sin<around*|(|h|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|sin<around*|(|x|)>*<around*|[|cos<around*|(|h|)>-1|]>|h>+lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|cos<around*|(|x|)>*sin<around*|(|h|)>|h>>>|<row|<cell|>|<cell|=>|<cell|sin<around*|(|x|)>*lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|cos<around*|(|h|)>-1|h>+cos<around*|(|x|)>*lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|sin<around*|(|h|)>|h>>>|<row|<cell|>|<cell|=>|<cell|cos<around*|(|x|)>.>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
<\wide-block>
|
||
|
|
<tformat|<cwith|1|1|2|2|cell-valign|c>|<cwith|1|1|1|1|cell-valign|c>|<table|<row|<\cell>
|
||
|
|
<with|gr-mode|<tuple|edit|line>|gr-frame|<tuple|scale|1.00003cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|0.4par|0.6par|center>|gr-grid|<tuple|cartesian|<point|0|0>|1>|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|4|none>>|gr-edit-grid|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-grid-aspect|<tuple|<tuple|axes|#808080>|<tuple|1|#c0c0c0>|<tuple|4|#e0e0ff>>|gr-grid-aspect-props|<tuple|<tuple|axes|#808080>|<tuple|1|#c0c0c0>|<tuple|4|#e0e0ff>>|gr-snap|<tuple|curve
|
||
|
|
point|curve-curve intersection>|gr-auto-crop|true|gr-snap-distance|5px|<graphics||<carc|<point|-1.25|1.25>|<point|1.25|-1.25>|<point|-1.25|-1.25>>|<line|<point|0|0>|<point|1.76776695296637|0.0>>|<math-at|A|<point|1.85482913922745|-0.111883894438878>>|<math-at|B|<point|0.8777543511513498|0.9301335317706395>>|<math-at|h|<point|0.5452616202957267|0.07332534071961085>>|<math-at|D|<point|1.7662881710946974|1.3363649726233786>>|<line|<point|1.45083|1.01>|<point|1.7677660586197912|-0.001778199046468032>>|<line|<point|0|0.00356307>|<point|1.7577628339396612|1.225535589607908>>|<line|<point|1.7577628339396612|1.225535589607908>|<point|1.7748135082497336|-0.010638297872340425>>|<math-at|O|<point|-0.4010941311028125|-0.29750995604183605>>>>
|
||
|
|
</cell>|<\cell>
|
||
|
|
First, assume <math|0\<less\>h\<less\><dfrac|\<pi\>|2>>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<around*|\||\<Delta\>O*A*B|\|>\<less\><around*|\||<wide|O*A*B|\<invbreve\>>|\|>\<less\><around*|\||\<Delta\>O*A*D|\|>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<dfrac|1|2>*sin<around*|(|h|)>\<less\><dfrac|1|2>*h\<less\><dfrac|1|2>*tan<around*|(|h|)>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
\<Rightarrow\><space|1em>*sin<around*|(|h|)>\<less\>h\<less\>tan<around*|(|h|)>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
\<Rightarrow\><space|1em>1\<less\><dfrac|h|sin
|
||
|
|
h>\<less\><dfrac|1|cos h>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
\<Rightarrow\><space|1em>cos h\<less\><dfrac|sin h|h>\<less\>1
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
\<Rightarrow\><space|1em>lim<rsub|h\<rightarrow\>0> cos
|
||
|
|
h\<leqslant\>lim<rsub|h\<rightarrow\>0> <dfrac|sin
|
||
|
|
h|h>\<leqslant\>lim<rsub|h\<rightarrow\>0> 1
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
\<Rightarrow\><space|1em>1\<leqslant\>lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|sin h|h>\<leqslant\>1.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
By the squeeze theorem, we have
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
lim<rsub|h\<rightarrow\>0> <dfrac|sin h|h>=1.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
\;
|
||
|
|
</equation*>
|
||
|
|
</cell>>>>
|
||
|
|
</wide-block>
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|cos<around*|(|h|)>-1|h>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|cos<around*|(|h|)>-1|h>*<dfrac|cos<around*|(|h|)>+1|cos<around*|(|h|)>+1>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|cos<rsup|2><around*|(|h|)>-1|h<around*|(|cos
|
||
|
|
h+1|)>>*>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|-sin<rsup|2> h|h<around*|(|cos
|
||
|
|
h+1|)>>*>>|<row|<cell|>|<cell|=>|<cell|-lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|sin<rsup|2> h|h>**lim<rsub|h\<rightarrow\>0> <dfrac|1|cos
|
||
|
|
h+1>>>|<row|<cell|>|<cell|=>|<cell|-<dfrac|1|2>*lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|sin<rsup|2> h|h>=-<dfrac|1|2>*lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|sin h|h>*lim<rsub|h\<rightarrow\>0> sin h=0.>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
For your record
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<marked|lim<rsub|x\<rightarrow\>0> <dfrac|sin
|
||
|
|
x|x>=1,<space|1em>lim<rsub|x\<rightarrow\>0> <dfrac|1-cos x|x>=0>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\theorem>
|
||
|
|
<dueto|Derivaties of <math|sin> and <math|cos>>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<around*|(|sin x|)><rprime|'>=cos x,<space|1em><around*|(|cos
|
||
|
|
x|)><rprime|'>=-sin x.
|
||
|
|
</equation*>
|
||
|
|
</theorem>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
Using the quotient rule, we can find the derivatives of other
|
||
|
|
trigonometric functions.
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<around*|(|tan x|)><rprime|'>=<around*|(|<dfrac|sin x|cos
|
||
|
|
x>|)><rprime|'>=<dfrac|cos<around*|(|x|)>*cos<around*|(|x|)>-sin<around*|(|x|)>*<around*|(|-sin
|
||
|
|
x|)>|cos<rsup|2> x>=<dfrac|1|cos<rsup|2> x>=sec<rsup|2> x.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\exercise>
|
||
|
|
Find the derivative of <math|cot x,sec x,csc x>.
|
||
|
|
</exercise>
|
||
|
|
|
||
|
|
<section|The Chain Rule>
|
||
|
|
|
||
|
|
Motivation: If <math|F<around*|(|x|)>=<sqrt|1+x<rsup|2>>>, then
|
||
|
|
<math|F<rprime|'><around*|(|x|)>=?>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\theorem>
|
||
|
|
<dueto|The Chain Rule>Suppose <math|g> is differentiable at <math|x>
|
||
|
|
and <math|f> is differentiable at <math|g<around*|(|x|)>>. Then
|
||
|
|
<math|f\<circ\>g> is differentiable at <math|x>, and
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<around*|(|f\<circ\>g|)><rprime|'><around*|(|x|)>=f<rprime|'><around*|(|g<around*|(|x|)>|)>*g<rprime|'><around*|(|x|)>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
Another way to say this. Suppose <math|y> is a function of <math|x>,
|
||
|
|
which is differentiable at <math|a>, and <math|z> is a function of
|
||
|
|
<math|y>, which is differentiable at <math|y<around*|(|a|)>>, then
|
||
|
|
<math|z> is a function of <math|x>, which is differentiable at
|
||
|
|
<math|a>, and
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<dfrac|d<hspace|0pt>z|d<hspace|0pt>x>=<dfrac|d<hspace|>z|d<hspace|0pt>y>*<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>
|
||
|
|
</equation*>
|
||
|
|
</theorem>
|
||
|
|
|
||
|
|
<\proof>
|
||
|
|
By the definition,
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|<around*|(|f\<circ\>g|)><rprime|'><around*|(|x|)>>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|f<around*|(|g<around*|(|x+h|)>|)>-f<around*|(|g<around*|(|x|)>|)>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|f<around*|(|g<around*|(|x+h|)>|)>-f<around*|(|g<around*|(|x|)>|)>|h>*<dfrac|g<around*|(|x+h|)>-g<around*|(|x|)>|g<around*|(|x+h|)>-g<around*|(|x|)>>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
|
||
|
|
<dfrac|f<around*|(|g<around*|(|x+h|)>|)>-f<around*|(|g<around*|(|x|)>|)>|g<around*|(|x+h|)>-g<around*|(|x|)>>**lim<rsub|h\<rightarrow\>0>
|
||
|
|
*<dfrac|g<around*|(|x+h|)>-g<around*|(|x|)>|h>>>|<row|<cell|>|<cell|=>|<cell|f<rprime|'><around*|(|g<around*|(|x|)>|)>*g<rprime|'><around*|(|x|)>.>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
\;
|
||
|
|
</proof>
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
Find <math|F<rprime|'><around*|(|x|)>> if
|
||
|
|
<math|F<around*|(|x|)>=<sqrt|1+x<rsup|2>>>.
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<strong|Answer:> <math|F=f\<circ\>g>, where
|
||
|
|
<math|f<around*|(|x|)>=<sqrt|x>,g<around*|(|x|)>=1+x<rsup|2>>.
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
F<rprime|'><around*|(|x|)>=f<rprime|'><around*|(|g<around*|(|x|)>|)>*g<rprime|'><around*|(|x|)>=<dfrac|1|2*<sqrt|g<around*|(|x|)>>>*2*x=<dfrac|1|<sqrt|1+x<rsup|2>>>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
Differentiate <math|y=<around*|(|x<rsup|3>-1|)><rsup|100>>.
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<strong|Answer:>\
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
y<rprime|'>=100*<around*|(|x<rsup|3>-1|)><rsup|99>*<around*|(|3*x<rsup|2>|)>=300*x<rsup|2>*<around*|(|x<rsup|3>-1|)><rsup|99>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<strong|Generalization>: For example,\
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<around*|(|f\<circ\>g\<circ\>h|)><rprime|'><around*|(|x|)>=f<rprime|'><around*|(|g<around*|(|h<around*|(|x|)>|)>|)>*g<rprime|'><around*|(|h<around*|(|x|)>|)>*h<rprime|'><around*|(|x|)>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
Find <math|f<rprime|'><around*|(|x|)>> if
|
||
|
|
<math|f<around*|(|x|)>=2<rsup|<sqrt|3+sin x>>>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
f<rprime|'><around*|(|x|)>=<around*|(|ln 2|)>*2<rsup|<sqrt|3+sin
|
||
|
|
x>>*<dfrac|cos x|2*<sqrt|3+sin x>>*.
|
||
|
|
</equation*>
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<section|Implicit differentiation>
|
||
|
|
|
||
|
|
Motivation: How to find <math|y<rprime|'><around*|(|x|)>> if
|
||
|
|
<math|e<rsup|x+y>=sin y>?
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
Find <math|<around*|\<nobracket\>|<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>|\|><rsub|x=3,y=4><rsub|>>
|
||
|
|
if <math|x<rsup|2>+y<rsup|2>=25>.
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<with|gr-mode|<tuple|edit|point>|gr-frame|<tuple|scale|1cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-grid|<tuple|cartesian|<point|0|0>|1>|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-auto-crop|true|<graphics||<line|<point|-4|0>|<point|4.6|0.0>>|<line|<point|0|-2.4>|<point|0.0|2.8>>|<carc|<point|-2|0>|<point|2.1|1.38777878078145e-17>|<point|2.0|0.7>>|<point|-2|0>|<point|2.1|6.93889e-18>|<point|1.40949|1.6>|<point|1.2|-1.6>>>
|
||
|
|
|
||
|
|
<strong|Answer:> Taking <math|<dfrac|d|d<hspace|0pt>x>> on both sides
|
||
|
|
of the equation (suppose <math|y> is a function of <math|x>):
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<dfrac|d|d<hspace|0pt>x>*<around*|(|x<rsup|2>+y<rsup|2>|)>=<dfrac|d|d<hspace|0pt>x>*<around*|(|25|)>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<dfrac|d|d<hspace|0pt>x>*x<rsup|2>+<dfrac|d|d<hspace|0pt>x>*y<rsup|2>=0
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
2*x+2*y*<with|color|blue|<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>>=0
|
||
|
|
<text|by the chain rule><space|1em>\<Rightarrow\><space|1em><dfrac|d<hspace|0pt>y|d<hspace|0pt>x>=-<dfrac|x|y>=-<dfrac|3|4>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
For this example, we can solve <math|y> for <math|x> and double check
|
||
|
|
the answer.
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
y=\<pm\><sqrt|25-x<rsup|2>>=<sqrt|25-x<rsup|2>>
|
||
|
|
near**<around*|(|3,4|)>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>=<dfrac|-2*x|2<sqrt|25-x<rsup|2>>>=-<dfrac|x|<sqrt|25-x<rsup|2>>>=-<dfrac|3|4>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
Lemniscate curve:
|
||
|
|
|
||
|
|
<\wide-tabular>
|
||
|
|
<tformat|<cwith|1|1|1|1|cell-halign|c>|<table|<row|<\cell>
|
||
|
|
<\script-output|asymptote|default>
|
||
|
|
% -width 0.5par
|
||
|
|
|
||
|
|
import contour;
|
||
|
|
|
||
|
|
size(10cm, 0);
|
||
|
|
|
||
|
|
draw((-3,0)--(3,0), arrow=Arrow, gray+linewidth(0.5pt));
|
||
|
|
|
||
|
|
label("$x$", (3,0), align=E);
|
||
|
|
|
||
|
|
draw((0,-1.5)--(0,1.5), arrow=Arrow, gray+linewidth(0.5pt));
|
||
|
|
|
||
|
|
label("$y$", (0,1.5), align=N);
|
||
|
|
|
||
|
|
real f(real x, real y) {return (x^2+y^2)^2-4*(x^2-y^2);}
|
||
|
|
|
||
|
|
guide[][] thegraphs = contour(f, a=(-2,-2), b=(2,2), new real[]
|
||
|
|
{0});
|
||
|
|
|
||
|
|
draw(thegraphs[0], linewidth(1pt));
|
||
|
|
|
||
|
|
label("$(x^2+y^2)^2=4(x^2-y^2)$", (0,1), align=E);
|
||
|
|
</script-output|<image|<tuple|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
|
||
|
|
|
||
|
|
\;
|
||
|
|
</cell>>>>
|
||
|
|
</wide-tabular>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<dfrac|d|d<hspace|0pt>x>*<around*|(|x<rsup|2>+y<rsup|2>|)><rsup|2>=<dfrac|d|d<hspace|0pt>x>
|
||
|
|
<around*|[|4*<around*|(|x<rsup|2>-y<rsup|2>|)>|]>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
2*<around*|(|x<rsup|2>+y<rsup|2>|)>*<around*|(|2*x+2*y*y<rprime|'>|)>=4*<around*|(|2x-2*y*y<rprime|'>|)>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
\<Rightarrow\><space|1em>y<rprime|'>=<dfrac|2x-x*<around*|(|x<rsup|2>+y<rsup|2>|)>|y*<around*|(|x<rsup|2>+y<rsup|2>|)>+2*y>
|
||
|
|
</equation*>
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<section|Derivatives of Logarithmic Functions>
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|y>|<cell|=>|<cell|log<rsub|b>
|
||
|
|
x>>|<row|<cell|x>|<cell|=>|<cell|b<rsup|y>>>|<row|<cell|<dfrac|d|d<hspace|0pt>x>
|
||
|
|
x>|<cell|=>|<cell|<dfrac|d|d<hspace|0pt>x>
|
||
|
|
b<rsup|y>>>|<row|<cell|1>|<cell|=>|<cell|<around*|(|ln
|
||
|
|
b|)>b<rsup|y>*y<rprime|'>>>|<row|<cell|\<Rightarrow\><space|1em>y<rprime|'>>|<cell|=>|<cell|<dfrac|1|<around*|(|ln
|
||
|
|
b|)>b<rsup|y>>>>|<row|<cell|>|<cell|=>|<cell|<dfrac|1|<around*|(|ln
|
||
|
|
b|)>*x>>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
<\theorem>
|
||
|
|
<dueto|Derivative of logarithmic functions>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<around*|(|log<rsub|b> x|)><rprime|'>=<dfrac|1|<around*|(|ln
|
||
|
|
b|)>*x>,<space|1em><around*|(|ln x|)><rprime|'>=<dfrac|1|x>.
|
||
|
|
</equation*>
|
||
|
|
</theorem>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<subsection|Logarithmic differentiation>
|
||
|
|
|
||
|
|
Using the properties
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
ln <around*|(|x*y|)>=ln x+ln y,<space|1em>ln x<rsup|y>=y*ln x
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
and implicit differentiation, one can simplify the differentiation of
|
||
|
|
some functions.
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
Find <math|<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>> if
|
||
|
|
<math|y=<dfrac|<around*|(|x<rsup|2>+1|)><rsup|5>*<around*|(|x-1|)><rsup|3>|<sqrt|x<rsup|2>+sin
|
||
|
|
x+2>>>.
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<strong|Answer:> First, take <math|ln> on both sides.
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|ln y>|<cell|=>|<cell|ln
|
||
|
|
<dfrac|<around*|(|x<rsup|2>+1|)><rsup|5>*<around*|(|x-1|)><rsup|3>|<sqrt|x<rsup|2>+sin
|
||
|
|
x+2>>>>|<row|<cell|>|<cell|=>|<cell|5*ln<around*|(|x<rsup|2>+1|)>+3*ln
|
||
|
|
<around*|(|x-1|)>-<dfrac|1|2>*ln <around*|(|x<rsup|2>+sin x+2|)>>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
Then take <math|<dfrac|d|d<hspace|0pt>x>> on both sides,
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|<dfrac|1|y>*y<rprime|'>>|<cell|=>|<cell|<dfrac|5|x<rsup|2>+1>\<cdot\>2**x+<dfrac|3|x-1>-<dfrac|1|2>\<cdot\><dfrac|2*x+cos
|
||
|
|
x|x<rsup|2>+sin x+2>>>|<row|<cell|>|<cell|=>|<cell|<dfrac|10*x|x<rsup|2>+1>+<dfrac|3|x-1>-<dfrac|2*x+cos
|
||
|
|
x|2*<around*|(|x<rsup|2>+sin x+2|)>>>>|<row|<cell|\<Rightarrow\>>|<cell|>|<cell|y<rprime|'>=y*<around*|(|<dfrac|10*x|x<rsup|2>+1>+<dfrac|3|x-1>-<dfrac|2*x+cos
|
||
|
|
x|2*<around*|(|x<rsup|2>+sin x+2|)>>|)>>>|<row|<cell|>|<cell|>|<cell|=<dfrac|<around*|(|x<rsup|2>+1|)><rsup|5>*<around*|(|x-1|)><rsup|3>|<sqrt|x<rsup|2>+sin
|
||
|
|
x+2>><around*|(|<dfrac|10*x|x<rsup|2>+1>+<dfrac|3|x-1>-<dfrac|2*x+cos
|
||
|
|
x|2*<around*|(|x<rsup|2>+sin x+2|)>>|)>>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
<\remark>
|
||
|
|
Let <math|z=ln y>, then
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<dfrac|d*z|d<hspace|0pt>x>=<dfrac|d*z|d<hspace|0pt>y>*<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>=<dfrac|1|y>*y<rprime|'>.
|
||
|
|
</equation*>
|
||
|
|
</remark>
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
<math|y=f<around*|(|x|)>=x<rsup|x>>.
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<strong|Answer:>\
|
||
|
|
|
||
|
|
<\eqnarray*>
|
||
|
|
<tformat|<table|<row|<cell|ln y>|<cell|=>|<cell|x*ln
|
||
|
|
x>>|<row|<cell|<dfrac|d|d<hspace|0pt>x> <around*|(|ln
|
||
|
|
y|)>>|<cell|=>|<cell|<dfrac|d|d<hspace|0pt>x> <around*|(|x*ln
|
||
|
|
x|)>>>|<row|<cell|<dfrac|y<rprime|'>|y>>|<cell|=>|<cell|ln
|
||
|
|
x+x*<dfrac|1|x>=1+ln x>>|<row|<cell|y<rprime|'>>|<cell|=>|<cell|x<rsup|x>*<around*|(|1+ln
|
||
|
|
x|)>>>>>
|
||
|
|
</eqnarray*>
|
||
|
|
|
||
|
|
<section|Rates of Change in the Economics and Social Sciences>
|
||
|
|
|
||
|
|
Read the slides.
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\section>
|
||
|
|
skip
|
||
|
|
</section>
|
||
|
|
|
||
|
|
<section|skip>
|
||
|
|
|
||
|
|
<section|Linear Approximations and Differentials>
|
||
|
|
|
||
|
|
<subsection|Linear approximations>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
Suppose <math|f> is differentiable at <math|a>, then
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
f<rprime|'><around*|(|a|)>=lim<rsub|x\<rightarrow\>a>
|
||
|
|
<dfrac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>\<approx\><dfrac|f<around*|(|x|)>-f<around*|(|a|)>|x-a><space|1em><text|when
|
||
|
|
>x <text|is close to >a
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
\<Rightarrow\><space|1em>f<around*|(|x|)>-f<around*|(|a|)>\<approx\>f<rprime|'><around*|(|a|)>*<around*|(|x-a|)>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
<marked|f<around*|(|x|)>\<approx\>f<around*|(|a|)>+f<rprime|'><around*|(|a|)>*<around*|(|x-a|)>>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
is called the <strong|linear approximation> of <math|f<around*|(|x|)>>.
|
||
|
|
And the linear function
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
L<around*|(|x|)>=f<around*|(|a|)>+f<rprime|'><around*|(|a|)>*<around*|(|x-a|)>
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
is called the <strong|linearization> of <math|f<around*|(|x|)>>.
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\wide-block>
|
||
|
|
<tformat|<cwith|1|-1|1|-1|cell-halign|l>|<cwith|1|-1|1|-1|cell-valign|t>|<cwith|1|-1|1|-1|cell-rsep|1spc>|<cwith|1|-1|1|-1|cell-bsep|1spc>|<cwith|1|-1|1|-1|cell-tsep|1spc>|<cwith|1|-1|1|-1|cell-lsep|1spc>|<table|<row|<\cell>
|
||
|
|
<draw-over|<image|<tuple|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
|
||
|
|
</cell>|<\cell>
|
||
|
|
<\example>
|
||
|
|
Estimate the value of <math|<sqrt|4.1>> and <math|<sqrt|3.99>>
|
||
|
|
using a linear approximation.
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<strong|Answer:> Let <math|f<around*|(|x|)>=<sqrt|x>>. Then
|
||
|
|
<math|f<rprime|'>=<dfrac|1|2*<sqrt|x>>>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
f<around*|(|4.1|)>\<approx\>f<around*|(|4|)>+f<rprime|'><around*|(|4|)>*0.1=2+<dfrac|0.1|4>=2.025.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
f<around*|(|3.99|)>\<approx\>f<around*|(|4|)>+f<rprime|'><around*|(|4|)>*<around*|(|-0.01|)>=2-<dfrac|0.01|4>=1.9975
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
\;
|
||
|
|
</cell>>>>
|
||
|
|
</wide-block>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
Approximate the value of <math|sin<around*|(|0.2|)>>.
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<strong|Answer:> Let <math|f<around*|(|x|)>=sin x>, then
|
||
|
|
<math|f<rprime|'><around*|(|x|)>=cos x>,\
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
f<around*|(|0.2|)>\<approx\>f<around*|(|0|)>+f<rprime|'><around*|(|0|)>*0.2=0+1\<times\>0.2=0.2
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\session|python|default>
|
||
|
|
<\output>
|
||
|
|
Python 3.9.6 [/Library/Developer/CommandLineTools/usr/bin/python3]\
|
||
|
|
|
||
|
|
Python plugin for TeXmacs.
|
||
|
|
|
||
|
|
Please see the documentation in Help -\<gtr\> Plugins -\<gtr\>
|
||
|
|
Python
|
||
|
|
</output>
|
||
|
|
|
||
|
|
<\unfolded-io>
|
||
|
|
\<gtr\>\<gtr\>\<gtr\>\
|
||
|
|
<|unfolded-io>
|
||
|
|
4.1**0.5
|
||
|
|
<|unfolded-io>
|
||
|
|
2.0248456731316584
|
||
|
|
</unfolded-io>
|
||
|
|
|
||
|
|
<\unfolded-io>
|
||
|
|
\<gtr\>\<gtr\>\<gtr\>\
|
||
|
|
<|unfolded-io>
|
||
|
|
3.99**0.5
|
||
|
|
<|unfolded-io>
|
||
|
|
1.997498435543818
|
||
|
|
</unfolded-io>
|
||
|
|
|
||
|
|
<\unfolded-io>
|
||
|
|
\<gtr\>\<gtr\>\<gtr\>\
|
||
|
|
<|unfolded-io>
|
||
|
|
math.sin(0.2)
|
||
|
|
<|unfolded-io>
|
||
|
|
0.19866933079506122
|
||
|
|
</unfolded-io>
|
||
|
|
|
||
|
|
<\input>
|
||
|
|
\<gtr\>\<gtr\>\<gtr\>\
|
||
|
|
<|input>
|
||
|
|
\;
|
||
|
|
</input>
|
||
|
|
</session>
|
||
|
|
|
||
|
|
\;
|
||
|
|
|
||
|
|
<\example>
|
||
|
|
Find an approximate value of <math|<sqrt|29|3>>.
|
||
|
|
</example>
|
||
|
|
|
||
|
|
<strong|Answer:> Let <math|f<around*|(|x|)>=<sqrt|x|3>>. Then
|
||
|
|
<math|f<rprime|'><around*|(|x|)>=<dfrac|1|3>*x<rsup|-<dfrac|2|3>>>.\
|
||
|
|
|
||
|
|
<\equation*>
|
||
|
|
f<around*|(|29|)>\<approx\>f*<around*|(|27|)>+f<rprime|'><around*|(|27|)>*<around*|(|29-27|)>=3+<dfrac|2|27>=<dfrac|83|27>.
|
||
|
|
</equation*>
|
||
|
|
|
||
|
|
<\session|python|default>
|
||
|
|
<\unfolded-io>
|
||
|
|
\<gtr\>\<gtr\>\<gtr\>\
|
||
|
|
<|unfolded-io>
|
||
|
|
29**(1/3)
|
||
|
|
<|unfolded-io>
|
||
|
|
3.072316825685847
|
||
|
|
</unfolded-io>
|
||
|
|
|
||
|
|
<\unfolded-io>
|
||
|
|
\<gtr\>\<gtr\>\<gtr\>\
|
||
|
|
<|unfolded-io>
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83/27
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<|unfolded-io>
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3.074074074074074
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</unfolded-io>
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</session>
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<subsection|Differentials>
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Suppose <math|y=f<around*|(|x|)>> is differentiable at <math|x>, then
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<\equation*>
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<dfrac|d<hspace|0pt>y|d<hspace|0pt>x>=f<rprime|'><around*|(|x|)><space|1em>\<Rightarrow\><space|1em>d<hspace|0pt>y=f<rprime|'><around*|(|x|)>*d<hspace|0pt>x,
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</equation*>
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where we call <math|d<hspace|0pt>x,d<hspace|0pt>y>
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<strong|differentials>.
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<\example>
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The radius of a sphere was measured and found to be
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<math|<rigid|21<space|0.6spc>cm>> with a possible error in
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measurement of at most <rigid|<math|0.05<space|0.6spc>cm>>. Estimate
|
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the maximum error in using this value of the radius to compute the
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volume of the sphere?
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</example>
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<strong|Answer:> Let <math|r,V> denote the radius and volume. Then
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<\equation*>
|
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V=<dfrac|4|3>*\<pi\>*r<rsup|3>.
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</equation*>
|
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Then <math|V<rprime|'>=4*\<pi\>*r<rsup|2>>
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<\equation*>
|
||
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|
d V=V<rprime|'><around*|(|r|)>*d*r=4*\<pi\>*r<rsup|2>*d*r
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</equation*>
|
||
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|
||
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Then
|
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||
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<\equation*>
|
||
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|
<around*|\||d*V|\|>=4*\<pi\>*r<rsup|2>*<around*|\||d*r|\|>\<leqslant\>4*\<pi\>*21<rsup|2>*0.05\<approx\>277*cm<rsup|3>.
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</equation*>
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||
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The relative change of the volume is
|
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|
||
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<\equation*>
|
||
|
|
<around*|\||<dfrac|d*V|V>|\|>=<dfrac|3|r><around*|\||d*r|\|>\<leqslant\><dfrac|0.05|7>\<approx\>0.007=7%.
|
||
|
|
</equation*>
|
||
|
|
</slide>
|
||
|
|
</slideshow>
|
||
|
|
</body>
|
||
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|
|
||
|
|
<\initial>
|
||
|
|
<\collection>
|
||
|
|
<associate|eqn-ver-sep|<macro|0fn>>
|
||
|
|
<associate|font|math=roman,Arial>
|
||
|
|
<associate|font-base-size|10>
|
||
|
|
<associate|font-family|rm>
|
||
|
|
<associate|font-series|medium>
|
||
|
|
<associate|info-flag|none>
|
||
|
|
<associate|item-vsep|<macro|0.4fn>>
|
||
|
|
<associate|magnification|1.2>
|
||
|
|
<associate|math-color|black>
|
||
|
|
<associate|ornament-border|0ln>
|
||
|
|
<associate|page-bot|25mm>
|
||
|
|
<associate|page-even|15mm>
|
||
|
|
<associate|page-even-footer|<htab|5mm><page-the-page><htab|5mm>>
|
||
|
|
<associate|page-even-header|>
|
||
|
|
<associate|page-height|auto>
|
||
|
|
<associate|page-medium|paper>
|
||
|
|
<associate|page-odd|15mm>
|
||
|
|
<associate|page-odd-footer|<htab|5mm><page-the-page><htab|5mm>>
|
||
|
|
<associate|page-odd-header|>
|
||
|
|
<associate|page-orientation|portrait>
|
||
|
|
<associate|page-right|15mm>
|
||
|
|
<associate|page-screen-margin|true>
|
||
|
|
<associate|page-top|15mm>
|
||
|
|
<associate|page-type|a4>
|
||
|
|
<associate|page-width|auto>
|
||
|
|
<associate|par-hyphen|normal>
|
||
|
|
<associate|par-par-sep|0.2fn>
|
||
|
|
<associate|preamble|false>
|
||
|
|
<associate|strong-color|black>
|
||
|
|
</collection>
|
||
|
|
</initial>
|
||
|
|
|
||
|
|
<\references>
|
||
|
|
<\collection>
|
||
|
|
<associate|auto-1|<tuple|1|1>>
|
||
|
|
<associate|auto-10|<tuple|4|5>>
|
||
|
|
<associate|auto-11|<tuple|5|6>>
|
||
|
|
<associate|auto-12|<tuple|6|8>>
|
||
|
|
<associate|auto-13|<tuple|6.1|8>>
|
||
|
|
<associate|auto-14|<tuple|7|9>>
|
||
|
|
<associate|auto-15|<tuple|8|9>>
|
||
|
|
<associate|auto-16|<tuple|9|9>>
|
||
|
|
<associate|auto-17|<tuple|10|9>>
|
||
|
|
<associate|auto-18|<tuple|10.1|9>>
|
||
|
|
<associate|auto-19|<tuple|10.2|10>>
|
||
|
|
<associate|auto-2|<tuple|1.1|1>>
|
||
|
|
<associate|auto-3|<tuple|1.2|2>>
|
||
|
|
<associate|auto-4|<tuple|1.3|2>>
|
||
|
|
<associate|auto-5|<tuple|2|3>>
|
||
|
|
<associate|auto-6|<tuple|2.1|3>>
|
||
|
|
<associate|auto-7|<tuple|2.2|4>>
|
||
|
|
<associate|auto-8|<tuple|3|4>>
|
||
|
|
<associate|auto-9|<tuple|3|4>>
|
||
|
|
</collection>
|
||
|
|
</references>
|
||
|
|
|
||
|
|
<\auxiliary>
|
||
|
|
<\collection>
|
||
|
|
<\associate|toc>
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Derivatives
|
||
|
|
of Polynomials and Exponential Functions>
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-1><vspace|0.5fn>
|
||
|
|
|
||
|
|
<with|par-left|<quote|1tab>|1.1<space|2spc>Power functions
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-2>>
|
||
|
|
|
||
|
|
<with|par-left|<quote|1tab>|1.2<space|2spc>Linear combination
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-3>>
|
||
|
|
|
||
|
|
<with|par-left|<quote|1tab>|1.3<space|2spc>Exponential functions
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-4>>
|
||
|
|
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>The
|
||
|
|
Product and Quotient Rules> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-5><vspace|0.5fn>
|
||
|
|
|
||
|
|
<with|par-left|<quote|1tab>|2.1<space|2spc>The product rule
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-6>>
|
||
|
|
|
||
|
|
<with|par-left|<quote|1tab>|2.2<space|2spc>The quotient rule
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-7>>
|
||
|
|
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>Derivatives
|
||
|
|
of Trigonometric Functions> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-8><vspace|0.5fn>
|
||
|
|
|
||
|
|
<with|par-left|<quote|1tab>|Derivative of
|
||
|
|
<with|color|<quote|black>|font-family|<quote|rm>|<with|math-display|<quote|true>|<with|mode|<quote|math>|<with|font-base-size|<quote|12>|sin
|
||
|
|
x>>>> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-9>>
|
||
|
|
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|4<space|2spc>The
|
||
|
|
Chain Rule> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-10><vspace|0.5fn>
|
||
|
|
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|5<space|2spc>Implicit
|
||
|
|
differentiation> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-11><vspace|0.5fn>
|
||
|
|
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|6<space|2spc>Derivatives
|
||
|
|
of Logarithmic Functions> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-12><vspace|0.5fn>
|
||
|
|
|
||
|
|
<with|par-left|<quote|1tab>|6.1<space|2spc>Logarithmic differentiation
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-13>>
|
||
|
|
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|7<space|2spc>Rates
|
||
|
|
of Change in the Economics and Social Sciences>
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-14><vspace|0.5fn>
|
||
|
|
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|8<space|2spc>skip>
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-15><vspace|0.5fn>
|
||
|
|
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|9<space|2spc>skip>
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-16><vspace|0.5fn>
|
||
|
|
|
||
|
|
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|10<space|2spc>Linear
|
||
|
|
Approximations and Differentials> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-17><vspace|0.5fn>
|
||
|
|
|
||
|
|
<with|par-left|<quote|1tab>|10.1<space|2spc>Linear approximations
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-18>>
|
||
|
|
|
||
|
|
<with|par-left|<quote|1tab>|10.2<space|2spc>Differentials
|
||
|
|
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
|
||
|
|
<no-break><pageref|auto-19>>
|
||
|
|
</associate>
|
||
|
|
</collection>
|
||
|
|
</auxiliary>
|