Chapter 3

In Problems 1-18, find \(D_{x} y\). $$ y=2 \sin x+3 \cos x $$

In Problems \(1-8\), find \(d^{3} y / d x^{3}\). $$ y=x^{3}+3 x^{2}+6 x $$

Show that \(f\) has an inverse by showing that it is strictly monotonic. \(f(x)=-x^{5}-x^{3}-x\)

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

In Problems \(1-4\), use the definition $$ f^{\prime}(c)=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h} $$ to find the indicated derivative. 1\. \(f^{\prime}(1)\) if \(f(x)=x^{2}\)

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

In Problems 1-20, find \(D_{x} y\). $$ y=(1+x)^{15} $$

Each edge of a variable cube is increasing at a rate of 3 inches per second. How fast is the volume of the cube increasing when an edge is 12 inches long?

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

In Problems \(1-8\), find \(d^{3} y / d x^{3}\). $$ y=x^{5}+x^{4} $$

Show that \(f\) has an inverse by showing that it is strictly monotonic. \(f(x)=x^{7}+x^{5}+x^{3}+x\)

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

\(f^{\prime}(2)\) if \(f(t)=(2 t)^{2}\)

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

In Problems 1-20, find \(D_{x} y\). $$ y=(7+x)^{5} $$

Assuming that a soap bubble retains its spherical shape as it expands, how fast is its radius increasing when its radius is 3 inches if air is blown into it at a rate of 3 cubic inches per second?

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

In Problems \(1-8\), find \(d^{3} y / d x^{3}\). $$ y=(3 x+5)^{3} $$

Show that \(f\) has an inverse by showing that it is strictly monotonic. \(f(\theta)=\cos \theta, 0 \leq \theta \leq \pi\)

Find \(D_{x} y\). $$ y=5 \sinh ^{2} x $$

\(f^{\prime}(3)\) if \(f(t)=t^{2}-t\)

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

In Problems 1-20, find \(D_{x} y\). $$ y=(3-2 x)^{5} $$

An airplane, flying horizontally at an altitude of 1 mile, passes directly over an observer. If the constant speed of the airplane is 400 miles per hour, how fast is its distance from the observer increasing 45 seconds later? Hint: Note that in 45 seconds \(\left(\frac{3}{4} \cdot \frac{1}{60}=\frac{1}{80}\right.\) hour \()\), the airplane goes 5 miles.

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

In Problems \(1-8\), find \(d^{3} y / d x^{3}\). $$ y=(3-5 x)^{5} $$

Show that \(f\) has an inverse by showing that it is strictly monotonic. \(f(x)=\cot x=\frac{\cos x}{\sin x}, 0

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

\(f^{\prime}(4)\) if \(f(s)=\frac{1}{s-1}\)

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

In Problems 1-20, find \(D_{x} y\). $$ y=\left(4+2 x^{2}\right)^{7} $$

A student is using a straw to drink from a conical paper cup, whose axis is vertical, at a rate of 3 cubic centimeters per second. If the height of the cup is 10 centimeters and the diameter of its opening is 6 centimeters, how fast is the level of the liquid falling when the depth of the liquid is 5 centimeters?

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

In Problems \(1-8\), find \(d^{3} y / d x^{3}\). $$ y=\sin (7 x) $$

$$ x y^{2}=x-8 $$

Find \(D_{x} y\). $$ y=\cosh (3 x+1) $$

In Problems 5-22, use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). 5\. \(s(x)=2 x+1\)

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

In Problems 1-20, find \(D_{x} y\). $$ y=\left(x^{3}-2 x^{2}+3 x+1\right)^{11} $$

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

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

$$ x^{2}+2 x^{2} y+3 x y=0 $$

Show that \(f\) has an inverse by showing that it is strictly monotonic. \(f(x)=x^{2}+x-6, x \geq 2\)

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

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

In Problems 1-20, find \(D_{x} y\). $$ y=\left(x^{2}-x+1\right)^{-7} $$

A woman on a dock is pulling in a rope fastened to the bow of a small boat. If the woman's hands are 10 feet higher than the point where the rope is attached to the boat and if she is retrieving the rope at a rate of 2 feet per second, how fast is the boat approaching the dock when 25 feet of rope is still out?

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

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

$$ 4 x^{3}+7 x y^{2}=2 y^{3} $$