Chapter 3

Find \(d y\). $$ y=x^{2}+x-3 $$

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

1-6, show that \(f\) has an inverse by showing that it is strictly monotonic. $$ f(x)=-x^{5}-x^{3}-x $$

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?

Assuming that each equation defines a differentiable function of \(x\), find \(D_{x} y\) by implicit differentiation. \(y^{2}-x^{2}=1\)

Find \(d^{3} y / d x^{3}\). $$ y=x^{3}+3 x^{2}+6 x $$

Find \(D_{x} y\). $$ y=(1+x)^{15} $$

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=2 \sin x+3 \cos x $$

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

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

Find \(d y\). $$ y=7 x^{3}+3 x^{2}+1 $$

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

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

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?

Assuming that each equation defines a differentiable function of \(x\), find \(D_{x} y\) by implicit differentiation. \(9 x^{2}+4 y^{2}=36\)

Find \(d^{3} y / d x^{3}\). $$ y=x^{5}+x^{4} $$

Find \(D_{x} y\). $$ y=(7+x)^{5} $$

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

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

Use the definition $$f^{\prime}(c)=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$$ to find the indicated derivative. $$ f^{\prime}(2) \text { if } f(t)=(2 t)^{2} $$

Find \(d y\). $$ y=(2 x+3)^{-4} $$

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

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

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.

Assuming that each equation defines a differentiable function of \(x\), find \(D_{x} y\) by implicit differentiation. \(x y=1\)

Find \(d^{3} y / d x^{3}\). $$ y=(3 x+5)^{3} $$

Find \(D_{x} y\). $$ y=(3-2 x)^{5} $$

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

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

Use the definition $$f^{\prime}(c)=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$$ to find the indicated derivative. $$ f^{\prime}(3) \text { if } f(t)=t^{2}-t $$

Find \(d y\). $$ y=\left(3 x^{2}+x+1\right)^{-2} $$

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

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

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?

Assuming that each equation defines a differentiable function of \(x\), find \(D_{x} y\) by implicit differentiation. \(x^{2}+\alpha^{2} y^{2}=4 \alpha^{2}\), where \(\alpha\) is a constant

Find \(d^{3} y / d x^{3}\). $$ y=(3-5 x)^{5} $$

Find \(D_{x} y\). $$ y=\left(4+2 x^{2}\right)^{7} $$

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

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

Use the definition $$f^{\prime}(c)=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$$ to find the indicated derivative. $$ f^{\prime}(4) \text { if } f(s)=\frac{1}{s-1} $$

Find \(d y\). $$ y=(\sin x+\cos x)^{3} $$

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

Assuming that each equation defines a differentiable function of \(x\), find \(D_{x} y\) by implicit differentiation. \(x y^{2}=x-8\)

Find \(d^{3} y / d x^{3}\). $$ y=\sin (7 x) $$

Find \(D_{x} y\). $$ y=\left(x^{3}-2 x^{2}+3 x+1\right)^{11} $$

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

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

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

Find \(d y\). $$ y=(\tan x+1)^{3} $$

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