mirror of https://gitee.com/XmacsLabs/planet.git
1101 lines
361 KiB
Tcl
1101 lines
361 KiB
Tcl
<TeXmacs|2.1.2>
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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>
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<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>
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<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>
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<dfrac|g<around*|(|x+h|)>*<around*|[|f<around*|(|x+h|)>-f<around*|(|x|)>|]>|h>+lim<rsub|h\<rightarrow\>0>
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<dfrac|f<around*|(|x|)>*<around*|[|g<around*|(|x+h|)>-g<around*|(|x|)>|]>|h>>>|<row|<cell|>|<cell|=>|<cell|lim<rsub|h\<rightarrow\>0>
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g<around*|(|x+h|)>*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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f<around*|(|x|)>*lim<rsub|h\<rightarrow\>0>
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<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|)>.>>>>
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</eqnarray*>
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\;
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</proof>
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Intuitive explanation
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<\wide-block>
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<tformat|<cwith|1|1|2|2|cell-valign|c>|<cwith|1|1|1|1|cell-valign|c>|<table|<row|<\cell>
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|png>|0.33par|||>
|
|
</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);
|
|
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|ps>|0.5par|||>>
|
|
|
|
\;
|
|
</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>
|
|
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|png>|0.4par|||>
|
|
</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>
|
|
83/27
|
|
<|unfolded-io>
|
|
3.074074074074074
|
|
</unfolded-io>
|
|
</session>
|
|
|
|
<subsection|Differentials>
|
|
|
|
Suppose <math|y=f<around*|(|x|)>> is differentiable at <math|x>, then
|
|
|
|
<\equation*>
|
|
<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,
|
|
</equation*>
|
|
|
|
where we call <math|d<hspace|0pt>x,d<hspace|0pt>y>
|
|
<strong|differentials>.
|
|
|
|
<\example>
|
|
The radius of a sphere was measured and found to be
|
|
<math|<rigid|21<space|0.6spc>cm>> with a possible error in
|
|
measurement of at most <rigid|<math|0.05<space|0.6spc>cm>>. Estimate
|
|
the maximum error in using this value of the radius to compute the
|
|
volume of the sphere?
|
|
</example>
|
|
|
|
<strong|Answer:> Let <math|r,V> denote the radius and volume. Then
|
|
|
|
<\equation*>
|
|
V=<dfrac|4|3>*\<pi\>*r<rsup|3>.
|
|
</equation*>
|
|
|
|
Then <math|V<rprime|'>=4*\<pi\>*r<rsup|2>>
|
|
|
|
<\equation*>
|
|
d V=V<rprime|'><around*|(|r|)>*d*r=4*\<pi\>*r<rsup|2>*d*r
|
|
</equation*>
|
|
|
|
Then
|
|
|
|
<\equation*>
|
|
<around*|\||d*V|\|>=4*\<pi\>*r<rsup|2>*<around*|\||d*r|\|>\<leqslant\>4*\<pi\>*21<rsup|2>*0.05\<approx\>277*cm<rsup|3>.
|
|
</equation*>
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The relative change of the volume is
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<\equation*>
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<around*|\||<dfrac|d*V|V>|\|>=<dfrac|3|r><around*|\||d*r|\|>\<leqslant\><dfrac|0.05|7>\<approx\>0.007=7%.
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</equation*>
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</slide>
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</slideshow>
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</body>
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<\auxiliary>
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<\collection>
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<\associate|toc>
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<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|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|<quote|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|<quote|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|<quote|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|<quote|bold>|math-font-series|<quote|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|<quote|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|<quote|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|<quote|bold>|math-font-series|<quote|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|<quote|1tab>|Derivative of
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<with|color|<quote|black>|font-family|<quote|rm>|<with|math-display|<quote|true>|<with|mode|<quote|math>|<with|font-base-size|<quote|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|<quote|bold>|math-font-series|<quote|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|<quote|bold>|math-font-series|<quote|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|<quote|bold>|math-font-series|<quote|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|<quote|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|<quote|bold>|math-font-series|<quote|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|<quote|bold>|math-font-series|<quote|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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<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|9<space|2spc>skip>
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<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|10<space|2spc>Linear
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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>>
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<with|par-left|<quote|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|<quote|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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