Chapter 3

Find the indicated derivative. \(D_{x} \ln \left(x^{2}+3 x+\pi\right)\)

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\pi x^{7}-2 x^{5}-5 x^{-2} $$

Suppose that an object moves along a coordinate line so that its directed distance from the origin after \(t\) seconds is \(\sqrt{2 t+1}\) feet. (a) Find its instantaneous velocity at \(t=\alpha, \alpha>0\). (b) When will it reach a velocity of \(\frac{1}{2}\) foot per second? (see Example 5.)

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

A light in a lighthouse 1 kilometer offshore from a straight shoreline is rotating at 2 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point \(\frac{1}{2}\) kilometer from the point opposite the lighthouse?

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

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

$$ y+\cos \left(x y^{2}\right)+3 x^{2}=4 ;(1,0) $$

Find the indicated derivative. \(D_{x} \ln \left(3 x^{3}+2 x\right)\)

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

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=x^{12}+5 x^{-2}-\pi x^{-10} $$

If a particle moves along a coordinate line so that its directed distance from the origin after \(t\) seconds is \(\left(-t^{2}+4 t\right)\) feet, when did the particle come to a momentary stop (i.e., when did its instantaneous velocity become zero)?

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

An aircraft spotter observes a plane flying at a constant altitude of 4000 feet toward a point directly above her head. She notes that when the angle of elevation is \(\frac{1}{2}\) radian it is increasing at a rate of \(\frac{1}{10}\) radian per second. What is the speed of the airplane?

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

Let \(n !=n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1\). Thus, \(\quad 4 !=\) \(4 \cdot 3 \cdot 2 \cdot 1=24\) and \(5 !=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\). We give \(n !\) the name \(\boldsymbol{n}\) factorial. Show that \(D_{x}^{n}\left(x^{n}\right)=n !\).

$$ x^{2 / 3}-y^{2 / 3}-2 y=2 ;(1,-1) $$

Find the indicated derivative. \(D_{x} \ln (x-4)^{3}\)

Find \(D_{x} y\). $$ y=\tanh ^{-1}(2 x-3) $$

\(G(x)=\frac{2 x-1}{x-4}\)

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

A certain bacterial culture is growing so that it has a mass of \(\frac{1}{2} t^{2}+1\) grams after \(t\) hours. C (a) How much did it grow during the interval \(2 \leq t \leq 2.01 ?\) (b) What was its average growth rate during the interval \(2 \leq t \leq 2.01 ?\) (c) What was its instantaneous growth rate at \(t=2 ?\)

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

Chris, who is 6 feet tall, is walking away from a street light pole 30 feet high at a rate of 2 feet per second. (a) How fast is his shadow increasing in length when Chris is 24 feet from the pole? 30 feet? (b) How fast is the tip of his shadow moving? (c) To follow the tip of his shadow, at what angular rate must Chris be lifting his eyes when his shadow is 6 feet long?

In Problems 1-18, find \(D_{x} y\). $$ y=\sec ^{3} x $$

Find a formula for $$ D_{x}^{n}\left(a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\right) $$

Find the indicated derivative. \(D_{x} \ln \sqrt{3 x-2}\)

Find \(D_{x} y\). $$ y=\operatorname{coth}^{-1}\left(x^{5}\right) $$

\(G(x)=\frac{2 x}{x^{2}-x}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=2 x^{-6}+x^{-1} $$

A business is prospering in such a way that its total (accumulated) profit after \(t\) years is \(1000 t^{2}\) dollars. (a) How much did the business make during the third year (between \(t=2\) and \(t=3\) )? (b) What was its average rate of profit during the first half of the third year, between \(t=2\) and \(t=2.5\) ? (The rate will be in dollars per year.) (c) What was its instantaneous rate of profit at \(t=2\) ?

In Problems 1-20, find \(D_{x} y\). $$ y=\left(2-3 x^{2}\right)^{4}\left(x^{7}+3\right)^{3} $$

The vertex angle \(\theta\) opposite the base of an isosceles triangle with equal sides of length 100 centimeters is increasing at \(\frac{1}{10}\) radian per minute. How fast is the area of the triangle increasing when the vertex angle measures \(\pi / 6\) radians? Hint: \(A=\frac{1}{2} a b \sin \theta\).

Find the equation of the tangent line to \(y=\cos x\) at \(x=1\).

Without doing any calculating, find each derivative. (a) \(D_{x}^{4}\left(3 x^{3}+2 x-19\right)\) (b) \(D_{x}^{12}\left(100 x^{11}-79 x^{10}\right)\) (c) \(D_{x}^{11}\left(x^{2}-3\right)^{5}\)

Find the indicated derivative. \(\frac{d y}{d x}\) if \(y=3 \ln x\)

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

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

A wire of length 8 centimeters is such that the mass between its left end and a point \(x\) centimeters to the right is \(x^{3}\) grams (Figure 12). Figure 12 (a) What is the average density of the middle 2 -centimeter segment of this wire? Note: Average density equals mass/length. (b) What is the actual density at the point 3 centimeters from the left end?

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

A long, level highway bridge passes over a railroad track that is 100 feet below it and at right angles to it. If an automobile traveling 45 miles per hour ( 66 feet per second) is directly above a train engine going 60 miles per hour ( 88 feet per second), how fast will they be separating 10 seconds later?

Find the equation of the tangent line to \(y=\cot x\) at \(x=\frac{\pi}{4}\).

Find a formula for \(D_{x}^{n}(1 / x)\).

Find the indicated derivative. \(\frac{d y}{d x}\) if \(y=x^{2} \ln x\)

Find \(D_{x} y\). $$ y=x^{2} \sinh ^{-1}\left(x^{5}\right) $$

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

Suppose that the revenue \(R(n)\) in dollars from producing \(n\) computers is given by \(R(n)=0.4 n-0.001 n^{2}\). Find the instantaneous rates of change of revenue when \(n=10\) and \(n=100\). (The instantaneous rate of change of revenue with respect to the amount of product produced is called the marginal revenue.)

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

Water is pumped at a uniform rate of 2 liters (1 liter \(=1000\) cubic centimeters) per minute into a tank shaped like a frustum of a right circular cone. The tank has altitude 80 centimeters and lower and upper radii of 20 and 40 centimeters, respectively (Figure 11). How fast is the water level rising when the depth of the water is 30 centimeters? Note: The volume, \(V\), of a frustum of a right circular cone of altitude \(h\) and lower and upper radii \(a\) and \(b\) is \(V=\frac{1}{3} \pi h \cdot\left(a^{2}+a b+b^{2}\right)\).