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

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

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

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

Find \(d y\). $$ y=\left(1-e^{x}\right) \ln x $$

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

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?

Assuming that each equation defines a differentiable function of \(x\), find \(D_{x} y\) by implicit differentiation. \(4 x^{3}+7 x y^{2}=2 y^{3}\)

Find \(d^{3} y / d x^{3}\). $$ y=\frac{1}{x-1} $$

Find \(D_{x} y\). $$ y=\frac{1}{(x+3)^{5}} $$

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

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ r(x)=3 x^{2}+4 $$

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

Consider \(y=x^{2}+1\). (a) Sketch its graph as carefully as you can. (b) Draw the tangent line at \((1,2)\). (c) Estimate the slope of this tangent line. (d) Calculate the slope of the secant line through \((1,2)\) and \(\left(1.01,(1.01)^{2}+1.0\right)\) (e) Find by the limit process (see Example 1) the slope of the tangent line at \((1,2)\).

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

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

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?

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

Find \(d^{3} y / d x^{3}\). $$ y=\frac{3 x}{1-x} $$

Find \(D_{x} y\). $$ y=\frac{1}{\left(3 x^{2}+x-3\right)^{9}} $$

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

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

Find \(D_{x} y\) 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. (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 \((2,7)\).

If \(s=\sqrt{\left(t^{2}-\cot t+2\right)^{3}}\), find \(d s\).

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

Assuming that each equation defines a differentiable function of \(x\), find \(D_{x} y\) by implicit differentiation. \(\sqrt{5 x y}+2 y=y^{2}+x y^{3}\)

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

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

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

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

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{100}{x^{5}} $$

Let \(y=f(x)=x^{3}\). Find the value of \(d y\) in each case. (a) \(x=0.5, d x=1\) (b) \(x=-1, d x=0.75\)

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

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.)

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

Find \(f^{\prime \prime}(2)\). $$ f(x)=5 x^{3}+2 x^{2}+x $$

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

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

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

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

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

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

Find \(f^{\prime \prime}(2)\). $$ f(t)=\frac{2}{t} $$

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

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

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ f(x)=x^{3}+2 x^{2}+1 $$