Chapter 2

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

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

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

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

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

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

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the deriva- tive of each of the following. (a) \(f(x)=3 x^{3}\) (b) \(f(x)=2 x^{5}+3 x\) (c) \(f(x)=\frac{1}{3 x}\) (d) \(f(x)=\frac{1}{3 x^{2}+2}\) (e) \(f(x)=\sqrt{3 x}\) (f) \(f(x)=\sin 3 x\) (g) \(f(x)=\sqrt{x^{2}+5}\) (h) \(f(x)=\cos \pi x\)

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

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

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

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

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

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

Find \(D_{x} y .\) \(y=\sin ^{2} x+\cos ^{2} x\)

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

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

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?

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

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

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

The given limit is a derivative, but of what function \(f\) and at what point? (a) \(\lim _{h \rightarrow 0} \frac{3(1+h)-3}{h}\) (b) \(\lim _{h \rightarrow 0} \frac{4(2+h)^{3}-4(2)^{3}}{h}\) (c) \(\lim _{\Delta x \rightarrow 0} \frac{\sqrt{(1+\Delta x)^{3}}-1}{\Delta x}\) (d) \(\lim _{\Delta x \rightarrow 0} \frac{\sin (\pi+\Delta x)}{\Delta x}\) (e) \(\lim _{t \rightarrow x} \frac{4 / t-4 / x}{t-x}\) (f) \(\lim _{t \rightarrow x} \frac{\sin 3 x-\sin 3 t}{t-x}\) (g) \(\lim _{h \rightarrow 0} \frac{\tan (\pi / 4+h)-1}{h}\) (h) \(\lim _{h \rightarrow 0}\left(\frac{1}{\sqrt{5+h}}-\frac{1}{\sqrt{5}}\right) \frac{1}{h}\)

Find \(D_{x} y .\) \(y=1-\cos ^{2} x\)

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

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

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

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

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

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

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

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

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

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

In Problems \(5-29\), find the indicated derivative by using the rules that we have developed. $$ D_{x}\left(3 x^{5}\right) $$

Find \(D_{x} y .\) \(y=\csc x=1 / \sin x\)

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

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

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

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?

Assuming that each equation defines a differentiable function of \(x\), find \(D_{x}\) y by implicit differentiation. $$ x^{2}+2 x^{2} y+3 x y=0 $$

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

Find \(D_{x} y .\) \(y=\left(x^{2}-x+1\right)^{-7}\)

In Problems \(5-29\), find the indicated derivative by using the rules that we have developed. $$ D_{x}\left(x^{3}-3 x^{2}+x^{-2}\right) $$