More About Derivatives

REVIEW EXERCISES
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In Problems 1–42, find the derivative of each function. When a, b, or n appear, they are constants.

$y=(ax+b)^{n}$

$an(ax+b)^{n-1}$

$y=\sqrt{2ax}$

$y=x\sqrt{1-x}$

$(1-x)^{1/2}-{x\over 2(1-x)^{1/2}}$

${y=\frac{1}{\sqrt{x^{2}+1}}}$

$y=(x^{2}+4)^{3/2}$

$3x(x^2+4)^{1/2}$

$F(x) =\frac{x^{2}}{\sqrt{x^{2}-1}}$

$z={\frac{\sqrt{2ax-x^{2}}}{x}}$

$-\dfrac{a}{x\sqrt{2ax-x^2}}$

$y=\sqrt{x}+\sqrt[3]{x}$

$y=(e^{x}-x)^{5x}$

$(e^x-x)^{5x}\left(5\,\ln(e^x-x)+{5x(e^x-1)\over e^x-x}\right)$

$\phi (x)={\frac{(x^{2}-a^{2})^{3/2}}{\sqrt{x+a}}}$

$f(x)={\frac{x^{2}}{(x-1)^{2}}}$

$-{2x\over(x-1)^3}$

$u= (b^{1/2}-x^{1/2}) ^{2}$

$y=x\sec (2x)$

$\sec(2x)+2x\sec(2x)\tan(2x)$

$u=\cos ^{3}x$

$y=\sqrt{a^{2}\sin \left( \frac{x}{a}\right)}$

$\dfrac{a\cos\frac{x}{a}}{2 \left(a^2\sin\frac{x}{a}\right)^{1/2}}$

$\phi (z)=\sqrt{1+\sin z}$

$u=\sin v-\frac{1}{3}\sin ^{3}v$

$\cos v-\sin^2v\cos v = \cos^3 v$

$y=\tan \sqrt{\frac{\pi }{x}}$

$y= (1.05) ^{x}$

$1.05^{x}\,\ln 1.05$

$v=\ln (y^{2}+1)$

$z=\ln (\sqrt{u^{2}+25} - u)$

$- \dfrac{1}{\sqrt{u^2+25}}$

$y=x^{2}+2^{x}$

$y=\ln [\sin (2x)]$

${2\cot(2x)}$

$f(x)=e^{-x}\sin (2x+\pi)$

$g(x)=\ln (x^{2}-2x)$

${2x-2\over x^2-2x}$

$y=\ln \frac{x^{2}+1}{x^{2}-1}$

$y=e^{-x}\ln x$

$e^{-x}\left(-\ln x+ {1\over x}\right)$

$w=\ln \!\big(\sqrt{x+7}-\sqrt{x}\big)$

$y=\frac{1}{12}\ln \left( \frac{x}{\sqrt{144-x^{2}}}\right)$

$\dfrac{12}{x(144-x^2)}$

$y=\ln (\tan ^{2}x)$

$f(x) =\frac{e^{x}(x^{2}+4) }{(x-2)}$

$\dfrac{e^x(x^2+4)}{(x-2)} \left[ 1+\dfrac{2x}{x^2+4} -\dfrac{1}{x-2} \right]$

$y=\sin^{-1} (x-1) + \sqrt{2x-x^{2}}$

$y=2\sqrt{x}-2\tan ^{-1}\sqrt{x}$

${\sqrt{x}\over 1+x}$

$y=4\tan ^{-1}\frac{x}{2}+x$

$y=\sin ^{-1}(2x-1)$

${1\over\sqrt{x-x^2}}$

$y=x^{2}\tan ^{-1}\frac{1}{x}$

$y=x\tan ^{-1}x-\ln \sqrt{1+x^{2}}$

$\tan^{-1}(x)$

$y=\sqrt{1-x^{2}}(\sin ^{-1}x)$

$y=\tanh \frac{x}{2}+\frac{2x}{4+x^{2}}$

${1\over 2}\hbox{sech}^2(x/2)+{8-2x^2\over (4+x^2)^2}$

$y=x\sinh x$

$y=\sqrt{\sinh x}$

$\dfrac{\hbox{cosh}x}{2(\sinh x)^{1/2}}$

$y=\sinh ^{-1}e^{x}$

In Problems 43–48, find $y'=\frac{dy}{dx}$ using implicit differentiation.

$x=y^{5}+y$

${1\over 5y^4+1}$

$x=\cos ^{5}y+\cos y$

$\ln x+\ln y=x\cos y$

${y(x\,\cos y-1) \over x(1+xy\,\sin y)}$

$\tan (xy)=x$

$y=x+\sin (xy)$

${1+y\,\cos(xy)\over 1-x\,\cos(xy)}$

$x=\ln (\csc y+\cot y)$

In Problems 49–52, find $y^\prime$ and $y^{\prime\prime}$ .

$xy+3y^{2}=10x$

$y'=\dfrac{10-y}{6 y+x}$ ; $y''=\dfrac{2 (y-10) (3 y+x+30)}{(6 y+x)^3}$

$y^{3}+y=x^{2}$

$xe^{y}=4x^{2}$

$y'=\dfrac{8x-e^y}{xe^y}$ ; $y''=\dfrac{-64 x^2+8 x e^{y}+e^{2 y}}{x^2e^{2y}}$

$\ln (x+y) =8x$

The function $f(x) =e^{2x}$ has an inverse function $g.$ Find $g^\prime (1)$ .

$\dfrac{1}{2}$

The function $f(x) =\sin x$ defined on the restricted domain $\left[ -\frac{\pi }{2},\,\frac{\pi }{2}\right]$ has an inverse function $g.$ Find $g^\prime \left(\frac{1}{2}\right)$ .

In Problems 55–56, express each limit in terms of the number $e$ .

$\lim\limits_{n\rightarrow \infty }\left( 1+\frac{2}{5n}\right) ^{n}$

$e^{2/5}$

$\lim\limits_{h\rightarrow 0} ( 1+3h) ^{2/h}$

In Problems 57 and 58, find the exact value of each expression.

$\sinh 0$

$0$

$\cosh ( \ln 3)$

In Problems 59 and 60, establish each identity.

$\sinh x+\cosh x=e^{x}$

See Student Solutions Manual.

$\tanh (x+y) =\frac{\tanh x+\tanh y}{1+\tanh x\tanh y}$

If $f(x)=\sqrt{1-\sin ^{2}x}$ , find the domain of $f^\prime$ .

$x\not=(2k-1)\dfrac{\pi}{2}$ , $k$ an integer

If $f(x)=x^{1/2}(x-2)^{3/2}$ for all $x ≥ 2$ , find the domain of $f^\prime$ .

Let $f$ be the function defined by $f(x)=\sqrt{1+6x}.$
1. What are the domain and the range of $f$ ?
2. Find the slope of the tangent line to the graph of $f$ at $x=4$ .
3. Find the $y$ -intercept of the tangent line to the graph of $f$ at $x=4$ .
4. Give the coordinates of the point on the graph of $f$ where the tangent line is parallel to the line $y=x+12$ .

Domain: $\left\{x|x\ge -\dfrac{1}{6}\right\}$ ; Range: $\{y|y\ge 0\}$
$\dfrac{3}{5}$
$\dfrac{13}{5}$
$\left(\dfrac{4}{3},3\right)$

Tangent and Normal Lines Find equations of the tangent and normal lines to the graph of $y=x\sqrt{x+(x-1)^{2}}$ at the point $(2,\,2 \sqrt{3})$ .

Find the differential $dy$ if $x^{3}+2y^{2}=x^{2}y$ .

$dy={2xy-3x^2\over 4y-x^2}\,dx$

Measurement Error If $p$ is the period of a pendulum of length $L$ , the acceleration due to gravity may be computed by the formula $g=\frac{(4\pi ^{2}L)}{p^{2}}$ . If $L$ is measured with negligible error, but a $2{\%}$ error may occur in the measurement of $p$ , what is the approximate percentage error in the computation of $g$ ?

Linear Approximation Find a linear approximation to $\hbox{$y=x+\ln x$ at $x=1$.}$

$L(x)=2x-1$

Measurement Error If the percentage error in measuring the edge of a cube is $5{\%}$ , what is the percentage error in computing its volume?

For the function $f(x) =\tan x$ :
1. Find the differential $dy$ and $\Delta y$ when $x=0.$
2. Compare $dy$ to $\Delta y$ when $x=0$ and (i) $\Delta x=0.5$ , (ii) $\Delta x=0.1$ , and (iii) $\Delta x=0.01$ .

$dy=dx$ ; $\Delta y=\tan(\Delta x)$
(b)
1. $dy=0.5$ , $\Delta y\approx 0.546$ ;
2. $dy=0.1$ , $\Delta y\approx 0.100$ ;
3. $dy=0.01$ , $\Delta y\approx 0.010$

For the function $f(x) =\ln x$ :
1. Find the differential $dy$ and $\Delta y$ when $x=1.$
2. Compare $dy$ to $\Delta y$ when $x=1$ and (i) $\Delta x=0.5$ , (ii) $\Delta x=0.1$ , and (iii) $\Delta x=0.01$ .

If $f(x)=(x^{2}+1)^{(2-3x)}$ , find $f^\prime (1)$ .

$-\dfrac{1}{2} (1+\ln 8)$

Find $\lim\limits_{x\rightarrow 2}\frac{\ln x-\ln 2}{x-2}$ .

Find $y^{\prime }$ at $x=\frac{\pi }{2}$ and $y=\pi$ if $x\sin y+y\cos x=0$ .

$-2$

In Problems 74–77, find the Taylor Polynomial $P_{n}( x)$ for $f$ at $x_{0}$ for the given $n$ and $x_{0}$ .

$f(x)=e^{2x};\quad n=4, x_{0}=3$

$f(x)=\tan x;\quad n=4, x_{0}=0$

$P_4(x)=\dfrac{x^3}{3}+x$

$f(x)=\frac{1}{1+x}; \quad n=4, x_{0}=1$

$f(x)=\ln x; n=6, x_{0}=2$

$P_6(x)=-\dfrac{1}{384} (x-2)^6+\dfrac{1}{160} (x-2)^5-\dfrac{1}{64} (x-2)^4+\dfrac{1}{24} (x-2)^3-\dfrac{1}{8} (x-2)^2+\dfrac{x-2}{2}+\ln 2$

In Problems 78 and 79, for each function:

(a) Use the Intermediate Value Theorem to confirm that a zero exists in the given interval.
(b) Use Newton’s Method to find $c_{3}$ , the third approximation to the real zero.

$f(x)=8x^{4}-2x^{2}+5x-1$ , interval: $(0,1)$ . Let $c_{1}=0.5.$

$f(x)=2-x+\sin x$ , interval: $\left(\frac{\pi }{2},\pi \right)$ . Let $c_{1}=\frac{\pi}{2}.$

(a) See Student Solutions Manual.
$2.568$

1. Use the Intermediate Value Theorem to confirm that the function $f(x) =2\cos x-e^{x}$ has a zero in the interval $(0,1)$ .
2. Use graphing technology with Newton’s Method to find $c_{5}$ , the fifth approximation to the real zero. Use the midpoint of the interval for the first approximation $c_{1}$ .

Tangent Line Find an equation of the tangent line to the graph of $4xy-y^{2}=3$ at the point $( 1, 3)$ .

$y=6x-3$

REVIEW EXERCISESPrinted Page 9999

REVIEW EXERCISES
Printed Page 9999