> <\body> <\hide-preamble> >>>>>> >> > > >> > > >>>>> >>>> |-0.3em|>|0em||0em|>>>> <\slideshow> <\slide> > <\table-of-contents|toc> Derivatives of Polynomials and Exponential Functions> |.>>>>|> Power functions |.>>>>|> > Linear combination |.>>>>|> > Exponential functions |.>>>>|> > The Product and Quotient Rules> |.>>>>|> The product rule |.>>>>|> > The quotient rule |.>>>>|> > Derivatives of Trigonometric Functions> |.>>>>|> >>> |.>>>>|> > The Chain Rule> |.>>>>|> Implicit differentiation> |.>>>>|> Derivatives of Logarithmic Functions> |.>>>>|> Logarithmic differentiation |.>>>>|> > Rates of Change in the Economics and Social Sciences> |.>>>>|> skip> |.>>>>|> skip> |.>>>>|> Linear Approximations and Differentials> |.>>>>|> Linear approximations |.>>>>|> > Differentials |.>>>>|> > <\itemize> If =c>, where is a constant. Then <\equation*> f=lim0> -f|h>=lim0> =0. If =x>, then <\equation*> f=lim0> -f|h>=lim0> -x|h>=1. If =x>, then <\equation*> f=lim0> -f|h>=lim0> -x|h> <\equation*> =lim0> |h>=lim0> =2*x. In general, if =x,n\\>, then <\equation*> f=n*x. Hint: use the binomial expansion <\equation*> =a+n*a*b+|2>*a*b+\+b=>|>>>>*a*b, where <\equation*> >|>>>>=!>. The above results can be generalized to =x,a\\>.\ <\equation*> =a*x>. \; From this result we can find the derivative of any polynomial. <\equation*> *f+c*g|]>=c*f+c*g>. <\example> If =x+5*x+6*x-12,>then <\equation*> f=100*x+15*x+6 Let =b,b\0>. Then <\eqnarray*> >||0> -f|h>>>|||0> -b|h>=lim0> *-1|)>|h>>>|||*lim0> -1|h>.>>>> In fact,\ <\equation*> lim0> -1|h>=f if it exists, and <\equation*> f=f*b. In fact,\ <\equation*> |)>=*b>. In particular, <\equation*> |)>=e>. <\question> Prove the limit exists. Hint: tranform the limit to the form containing the term \>|)>>. to see a proof that the limit of the sequence |)>>exists as \>. \; <\definition> The of a function > is <\equation*> |f>, and the is <\equation*> 100*|f>. \; \; Motivation: How to find the derivative function of =x*e>? \; <\theorem> If and are both differentiable at , then is differentiable at . Moreover, <\equation*> =f*g+f*g. <\proof> From the definition, <\eqnarray*> >||0> *g-f*g|h>>>|||0> *g*g+f*g>-f*g|h>>>|||0> *-f|]>+f*-g|]>|h>>>|||0> *-f|]>|h>+lim0> *-g|]>|h>>>|||0> g*lim0> -f|h>+lim0> f*lim0> -g|h>>>|||*f+f*g.>>>> \; Intuitive explanation <\wide-block> || |png>|0.33par|||> |<\cell> Let ,v=g>. For small change x>, let u,\v> be the change in , respectively. Let > be the change in . <\eqnarray*> >||u|)>*v|)>-u*v>>|||v+v*\u+\u*\v-u*v>>|||v+v*\u+\u*\v>>>> <\eqnarray*> |\x>>||v+v*\u+\u*\v|\x>>>|||v|\x>+v*u|\x>+\u*v|\x>>>||>|+v*u+0as\x\0>>|||+v*u>>>> >>> \; <\example> \; <\enumerate-alpha> If =x*e>, find >.\ <\equation*> |)>=x*e+x*|)>=e+x*e=e*. Find the th derivative, >>. <\eqnarray*> |)>>||*|]>>>|||+e=e*.>>>> Repeat this process one more time, we get\ <\equation*> |)>=e*. In general <\equation*> f>=e*. <\exercise> Prove the general result using mathematical induction. <\theorem> If and are both differentiable at and \0>, then is differentiable at . Moreover, <\equation*> |)>=*g-f*g|g>. \; <\exercise> Prove the quotient rule using the definition of derivative. <\example> Let +x-2|x+6>>. Then <\eqnarray*> >||+x-2|)>*+6|)>-+x-2|)>*+6|)>|+6|)>>>>|||*+6|)>-+x-2|)>*|)>|+6|)>>>>|||+x+12*x+6-+3*x-6*x|)>|+6|)>>>>|||-2*x+6*x+12*x+6|+6|)>>.>>>> \; > By the definition <\eqnarray*> >||0> -sin|h>>>|||0> *cos+cos*sin-sin|h>>>|||0> *-1|]>+cos*sin|h>>>|||0> *-1|]>|h>+lim0> *sin|h>>>|||*lim0> -1|h>+cos*lim0> |h>>>|||.>>>> <\wide-block> || |gr-frame|>|gr-geometry||gr-grid||1>|gr-grid-old||1>|gr-edit-grid-aspect|||>|gr-edit-grid||1>|gr-edit-grid-old||1>|gr-grid-aspect|||>|gr-grid-aspect-props|||>|gr-snap||gr-auto-crop|true|gr-snap-distance|5px|||>||>|>|>|>|>||>||>||>|>>> |<\cell> First, assume h\|2>> <\equation*> O*A*B|\|>\>|\|>\O*A*D|\|> <\equation*> *sin\*h\*tan <\equation*> \*sin\h\tan <\equation*> \1\\ <\equation*> \cos h\\1 <\equation*> \lim0> cos h\lim0> \lim0> 1 <\equation*> \1\lim0> \1. By the squeeze theorem, we have <\equation*> lim0> =1. <\equation*> \; >>> <\eqnarray*> 0> -1|h>>||0> -1|h>*+1|cos+1>>>|||0> -1|h>*>>|||0> h|h>*>>|||0> h|h>**lim0> >>|||*lim0> h|h>=-*lim0> *lim0> sin h=0.>>>> For your record <\equation*> 0> =1,lim0> =0>. <\theorem> and > <\equation*> =cos x,=-sin x. \; Using the quotient rule, we can find the derivatives of other trigonometric functions. <\equation*> =|)>=*cos-sin*|cos x>= x>=sec x. <\exercise> Find the derivative of . Motivation: If =>>, then =?> \; <\theorem> Suppose is differentiable at and is differentiable at >. Then g> is differentiable at , and <\equation*> g|)>=f|)>*g. Another way to say this. Suppose is a function of , which is differentiable at , and is a function of , which is differentiable at >, then is a function of , which is differentiable at , and <\equation*> z|dx>=z|dy>*y|dx> <\proof> By the definition, <\eqnarray*> g|)>>||0> |)>-f|)>|h>>>|||0> |)>-f|)>|h>*-g|g-g>>>|||0> |)>-f|)>|g-g>**lim0> *-g|h>>>||||)>*g.>>>> \; <\example> Find > if =>>. g>, where =,g=1+x>. <\equation*> F=f|)>*g=>>*2*x=>>. \; <\example> Differentiate -1|)>>. \ <\equation*> y=100*-1|)>*|)>=300*x*-1|)>. \; : For example,\ <\equation*> g\h|)>=f|)>|)>*g|)>*h. <\example> Find > if =2>> <\equation*> f=*2>*>*. Motivation: How to find > if =sin y>? <\example> Find |y|dx>|\|>> if +y=25>. |gr-frame|>|gr-geometry||gr-grid||1>|gr-grid-old||1>|gr-edit-grid-aspect|||>|gr-edit-grid||1>|gr-edit-grid-old||1>|gr-auto-crop|true||>||>|||>||||>> Taking x>> on both sides of the equation (suppose is a function of ): <\equation*> x>*+y|)>=x>* <\equation*> x>*x+x>*y=0 <\equation*> 2*x+2*y*y|dx>>=0 \y|dx>=-=-. For this example, we can solve for and double check the answer. <\equation*> y=\>=> near** <\equation*> y|dx>=>>=->>=-. \; <\example> Lemniscate curve: <\wide-tabular> | <\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); |ps>|0.5par|||>> \; >>> <\equation*> x>*+y|)>=x> -y|)>|]> <\equation*> 2*+y|)>*|)>=4*|)> <\equation*> \y=+y|)>|y*+y|)>+2*y> <\eqnarray*> || x>>|||>>|x> x>||x> b>>|||b*y>>|y>||b>>>|||*x>>>>> <\theorem> <\equation*> x|)>=*x>,=. \; Using the properties <\equation*> ln =ln x+ln y,ln x=y*ln x and implicit differentiation, one can simplify the differentiation of some functions. <\example> Find y|dx>> if +1|)>*|+sin x+2>>>. First, take on both sides. <\eqnarray*> ||+1|)>*|+sin x+2>>>>|||+1|)>+3*ln -*ln +sin x+2|)>>>>> Then take x>> on both sides, \; <\eqnarray*> *y>||+1>\2**x+-\+sin x+2>>>|||+1>+-+sin x+2|)>>>>|>||=y*+1>+-+sin x+2|)>>|)>>>|||+1|)>*|+sin x+2>>+1>+-+sin x+2|)>>|)>>>>> <\remark> Let , then <\equation*> x>=y>*y|dx>=*y. <\example> =x>. \ <\eqnarray*> ||>|x> >||x> >>||y>>||=1+ln x>>|>||*>>>> Read the slides. \; <\section> skip \; Suppose is differentiable at , then <\equation*> f=lima> -f|x-a>\-f|x-a>x a <\equation*> \f-f\f* <\equation*> \f+f*>. is called the of >. And the linear function <\equation*> L=f+f* is called the of >. \; <\wide-block> |||||| |png>|0.4par|||>||gr-color||gr-frame|>||>>>>|0cm> |<\cell> <\example> Estimate the value of > and > using a linear approximation. Let =>. Then =>> <\equation*> f\f+f*0.1=2+=2.025. <\equation*> f\f+f*=2-=1.9975 \; >>> \; <\example> Approximate the value of >. Let =sin x>, then =cos x>,\ <\equation*> f\f+f*0.2=0+1\0.2=0.2 <\session|python|default> <\output> Python 3.9.6 [/Library/Developer/CommandLineTools/usr/bin/python3]\ Python plugin for TeXmacs. Please see the documentation in Help -\ Plugins -\ Python <\unfolded-io> \\\\ <|unfolded-io> 4.1**0.5 <|unfolded-io> 2.0248456731316584 <\unfolded-io> \\\\ <|unfolded-io> 3.99**0.5 <|unfolded-io> 1.997498435543818 <\unfolded-io> \\\\ <|unfolded-io> math.sin(0.2) <|unfolded-io> 0.19866933079506122 <\input> \\\\ <|input> \; \; <\example> Find an approximate value of >. Let =>. Then =*x>>.\ <\equation*> f\f*+f*=3+=. <\session|python|default> <\unfolded-io> \\\\ <|unfolded-io> 29**(1/3) <|unfolded-io> 3.072316825685847 <\unfolded-io> \\\\ <|unfolded-io> 83/27 <|unfolded-io> 3.074074074074074 Suppose > is differentiable at , then <\equation*> y|dx>=f\dy=f*dx, where we call x,dy> . <\example> The radius of a sphere was measured and found to be cm>> with a possible error in measurement of at most cm>>. Estimate the maximum error in using this value of the radius to compute the volume of the sphere? Let denote the radius and volume. Then <\equation*> V=*\*r. Then =4*\*r> <\equation*> d V=V*d*r=4*\*r*d*r Then <\equation*> =4*\*r*\4*\*21*0.05\277*cm. The relative change of the volume is <\equation*> |\|>=\\0.007=7%. <\initial> <\collection> > > > > <\references> <\collection> > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|toc> |math-font-series||1Derivatives of Polynomials and Exponential Functions> |.>>>>|> |1.1Power functions |.>>>>|> > |1.2Linear combination |.>>>>|> > |1.3Exponential functions |.>>>>|> > |math-font-series||2The Product and Quotient Rules> |.>>>>|> |2.1The product rule |.>>>>|> > |2.2The quotient rule |.>>>>|> > |math-font-series||3Derivatives of Trigonometric Functions> |.>>>>|> |Derivative of |font-family|||||sin x>>>> |.>>>>|> > |math-font-series||4The Chain Rule> |.>>>>|> |math-font-series||5Implicit differentiation> |.>>>>|> |math-font-series||6Derivatives of Logarithmic Functions> |.>>>>|> |6.1Logarithmic differentiation |.>>>>|> > |math-font-series||7Rates of Change in the Economics and Social Sciences> |.>>>>|> |math-font-series||8skip> |.>>>>|> |math-font-series||9skip> |.>>>>|> |math-font-series||10Linear Approximations and Differentials> |.>>>>|> |10.1Linear approximations |.>>>>|> > |10.2Differentials |.>>>>|> >