Chapter 3

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{\pi}{x} $$

In Problems 1-20, find \(D_{x} y\). $$ y=(3 x-2)^{2}\left(3-x^{2}\right)^{2} $$

A 20-foot ladder is leaning against a building. If the bottom of the ladder is sliding along the level pavement directly away from the building at 1 foot per second, how fast is the top of the ladder moving down when the foot of the ladder is 5 feet from the wall?

In Problems 1-18, find \(D_{x} y\). $$ y=\cot x=\frac{\cos x}{\sin x} $$

In Problems \(1-8\), find \(d^{3} y / d x^{3}\). $$ y=\frac{3 x}{1-x} $$

$$ x^{2} y=1+y^{2} x $$

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{\alpha}{x^{3}} $$

Consider \(y=x^{3}-1\). (a) Sketch its graph as carefully as you can. (b) Draw the tangent line at \((2,7)\). (c) Estimate the slope of this tangent line. C (d) Calculate the slope of the secant line through \((2,7)\) and \(\left(2.01,(2.01)^{3}-1.0\right)\) (e) Find by the limit process (see Example 1) the slope of the tangent line at \((2,7)\).

In Problems 1-20, find \(D_{x} y\). $$ y=\frac{1}{\left(3 x^{2}+x-3\right)^{9}} $$

We assume that an oil spill is being cleaned up by deploying bacteria that consume the oil at 4 cubic feet per hour. The oil spill itself is modeled in the form of a very thin cyclinder whose height is the thickness of the oil slick. When the thickness of the slick is \(0.001\) foot, the cylinder is 500 feet in diameter. If the height is decreasing at \(0.0005\) foot per hour, at what rate is the area of the slick changing?

In Problems 1-18, find \(D_{x} y\). $$ y=\frac{\sin x+\cos x}{\cos x} $$

In Problems 9-16, find \(f^{\prime \prime}(2)\). $$ f(x)=x^{2}+1 $$

Find \(D_{x} y\). $$ y=x^{2} \cosh x $$

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{100}{x^{5}} $$

Find the slopes of the tangent lines to the curve \(y=x^{2}-1\) at the points where \(x=-2,-1,0,1,2\) (see Example 2).

In Problems 1-20, find \(D_{x} y\). $$ y=\sin \left(x^{2}+x\right) $$

In Problems 1-18, find \(D_{x} y\). $$ y=\frac{\sin x+\cos x}{\tan x} $$

In Problems 9-16, find \(f^{\prime \prime}(2)\). $$ f(x)=5 x^{3}+2 x^{2}+x $$

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{3 \alpha}{4 x^{5}} $$

Find the slopes of the tangent lines to the curve \(y=x^{3}-3 x\) at the points where \(x=-2,-1,0,1,2\).

In Problems 1-20, find \(D_{x} y\). $$ y=\cos \left(3 x^{2}-2 x\right) $$

A child is flying a kite. If the kite is 90 feet above the child's hand level and the wind is blowing it on a horizontal course at 5 feet per second, how fast is the child paying out cord when 150 feet of cord is out? (Assume that the cord remains straight from hand to kite, actually an unrealistic assumption.)

In Problems 1-18, find \(D_{x} y\). $$ y=\sin x \cos x $$

In Problems 9-16, find \(f^{\prime \prime}(2)\). $$ f(t)=\frac{2}{t} $$

Find \(D_{x} y\). $$ y=\cosh 3 x \sinh x $$

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=x^{2}+2 x $$

In Problems 1-20, find \(D_{x} y\). $$ y=\cos ^{3} x $$

In Problems 1-18, find \(D_{x} y\). $$ y=\sin x \tan x $$

In Problems 9-16, find \(f^{\prime \prime}(2)\). $$ f(u)=\frac{2 u^{2}}{5-u} $$

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=3 x^{4}+x^{3} $$

In Problems 1-20, find \(D_{x} y\). $$ y=\sin ^{4}\left(3 x^{2}\right) $$

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 \(x=3\) ?

In Problems 1-18, find \(D_{x} y\). $$ y=\frac{\sin x}{x} $$

In Problems 9-16, find \(f^{\prime \prime}(2)\). $$ f(\theta)=(\cos \theta \pi)^{-2} $$

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=x^{4}+x^{3}+x^{2}+x+1 $$

In Problems 1-20, find \(D_{x} y\). $$ y=\left(\frac{x+1}{x-1}\right)^{3} $$

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?

In Problems 1-18, find \(D_{x} y\). $$ y=\frac{1-\cos x}{x} $$

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=3 x^{4}-2 x^{3}-5 x^{2}+\pi x+\pi^{2} $$

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 \(2 \leq t \leq 3\) ? C (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 \(t=2\).

In Problems 1-20, find \(D_{x} y\). $$ y=\left(\frac{x-2}{x-\pi}\right)^{-3} $$

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 \(9: 00 \mathrm{~A} . \mathrm{M}\). 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.

In Problems 1-18, find \(D_{x} y\). $$ y=x^{2} \cos x $$

In Problems 9-16, find \(f^{\prime \prime}(2)\). $$ f(s)=s\left(1-s^{2}\right)^{3} $$