Chapter 3

Find \(D_{x} y\) using the rules of this section. $$ y=x^{2}+2 x $$

Let \(y=1 / x\). Find the value of \(d y\) in each case. (a) \(x=1, d x=0.5\) (b) \(x=-2, d x=0.75\)

Find \(D_{x} y\). $$ y=\sinh x \cosh 4 x $$

A particle \(P\) is moving along the graph of \(y=\) \(\sqrt{x^{2}-4}, x \geq 2\), so that the \(x\) -coordinate of \(P\) is increasing at the rate of 5 units per second. How fast is the \(y\) -coordinate of \(P\) increasing when \(x=3\) ?

Assuming that each equation defines a differentiable function of \(x\), find \(D_{x} y\) by implicit differentiation. \(\cos \left(x y^{2}\right)=y^{2}+x\)

Find \(f^{\prime \prime}(2)\). $$ f(u)=\frac{2 u^{2}}{5-u} $$

Find \(D_{x} y\). $$ y=\sin ^{4}\left(3 x^{2}\right) $$

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\sin x \tan x $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ g(x)=x^{4}+x^{2} $$

Find \(D_{x} y\) using the rules of this section. $$ y=3 x^{4}+x^{3} $$

Find the equation of the tangent line to \(y=1 /(x-1)\) at \((0,-1)\)

Find \(D_{x} y\). $$ y=\tanh x \sinh 2 x $$

A metal disk expands during heating. If its radius increases at the rate of \(0.02\) inch per second, how fast is the area of one of its faces increasing when its radius is \(8.1\) inches?

Find \(D_{x} y\). $$ y=\left(\frac{x+1}{x-1}\right)^{3} $$

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\frac{\sin x}{x} $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ h(x)=\frac{2}{x} $$

Find \(D_{x} y\) using the rules of this section. $$ y=x^{4}+x^{3}+x^{2}+x+1 $$

Find \(D_{x} y\). $$ y=\operatorname{coth} 4 x \sinh x $$

Two ships sail from the same island port, one going north at 24 knots ( 24 nautical miles per hour) and the other east at 30 knots. The northbound ship departed at \(9: 00\) A.M. and the eastbound ship left at 11:00 A.M. How fast is the distance between them increasing at 2:00 P.M.? Hint: Let \(t=0\) at 11:00 A.M.

Find \(f^{\prime \prime}(2)\). $$ f(t)=t \sin (\pi / t) $$

Find \(D_{x} y\). $$ y=\left(\frac{x-2}{x-\pi}\right)^{-3} $$

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\frac{1-\cos x}{x} $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ S(x)=\frac{1}{x+1} $$

Find \(D_{x} y\) using the rules of this section. $$ y=3 x^{4}-2 x^{3}-5 x^{2}+\pi x+\pi^{2} $$

An object travels along a line so that its position \(s\) is \(s=t^{2}+1\) meters after \(t\) seconds. (a) What is its average velocity on the interval \(2 \leq t \leq 3 ?\) (b) What is its average velocity on the interval \(2 \leq t \leq 2.003\) ? (c) What is its average velocity on the interval \(2 \leq t \leq 2+h ?\) (d) Find its instantaneous velocity at \(t=2 .\)

Find \(D_{x} y\). $$ y=\sinh ^{-1}\left(x^{2}\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} \ln \left(x^{2}+3 x+\pi\right) $$

A light in a lighthouse 1 kilometer offshore from a straight shoreline is rotating at 2 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point \(\frac{1}{2}\) kilometer from the point opposite the lighthouse?

Find \(f^{\prime \prime}(2)\). $$ f(s)=s\left(1-s^{2}\right)^{3} $$

Find \(D_{x} y\). $$ y=\cos \left(\frac{3 x^{2}}{x+2}\right) $$

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=x^{2} \cos x $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ F(x)=\frac{6}{x^{2}+1} $$

Find \(D_{x} y\) using the rules of this section. $$ y=\pi x^{7}-2 x^{5}-5 x^{-2} $$

Suppose that an object moves along a coordinate line so that its directed distance from the origin after \(t\) seconds is \(\sqrt{2 t+1}\) feet. (a) Find its instantaneous velocity at \(t=\alpha, \alpha>0\). (b) When will it reach a velocity of \(\frac{1}{2}\) foot per second? (see Example 5.)

If \(y=x^{2}-3\), find the values of \(\Delta y\) and \(d y\) in each case. (a) \(x=2\) and \(d x=\Delta x=0.5\) (b) \(x=3\) and \(d x=\Delta x=-0.12\)

Find \(D_{x} y\). $$ y=\cosh ^{-1}\left(x^{3}\right) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x} \ln \left(3 x^{3}+2 x\right) $$

An aircraft spotter observes a plane flying at a constant altitude of 4000 feet toward a point directly above her head. She notes that when the angle of elevation is \(\frac{1}{2}\) radian it is increasing at a rate of \(\frac{1}{10}\) radian per second. What is the speed of the airplane?

Find \(f^{\prime \prime}(2)\). $$ f(x)=\frac{(x+1)^{2}}{x-1} $$

Find \(D_{x} y\). $$ y=\cos ^{3}\left(\frac{x^{2}}{1-x}\right) $$

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\frac{x \cos x+\sin x}{x^{2}+1} $$

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ F(x)=\frac{x-1}{x+1} $$

Find \(D_{x} y\) using the rules of this section. $$ y=x^{12}+5 x^{-2}-\pi x^{-10} $$

If a particle moves along a coordinate line so that its directed distance from the origin after \(t\) seconds is \(\left(-t^{2}+4 t\right)\) feet, when did the particle come to a momentary stop (i.e., when did its instantaneous velocity become zero)?

If \(y=x^{4}+2 x\), find the values of \(\Delta y\) and \(d y\) in each case. (a) \(x=2\) and \(d x=\Delta x=1\) (b) \(x=2\) and \(d x=\Delta x=0.005\)

Find \(D_{x} y\). $$ y=\tanh ^{-1}(2 x-3) $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$$$ D_{x} \ln (x-4)^{3} $$

Chris, who is 6 feet tall, is walking away from a street light pole 30 feet high at a rate of 2 feet per second. (a) How fast is his shadow increasing in length when Chris is 24 feet from the pole? 30 feet? (b) How fast is the tip of his shadow moving? (c) To follow the tip of his shadow, at what angular rate must Chris be lifting his eyes when his shadow is 6 feet long?

Find \(D_{x} y\). $$ y=(3 x-2)^{2}\left(3-x^{2}\right)^{2} $$

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\tan ^{2} x $$