Chapter 3

Use the trigonometric identity \(\sin 2 x=2 \sin x \cos x\) along with the Product Rule to find \(D_{x} \sin 2 x\).

If \(f(x)=x^{3}+3 x^{2}-45 x-6\), find the value of \(f^{\prime \prime}\) at each zero of \(f^{\prime}\), that is, at each point \(c\) where \(f^{\prime}(c)=0\).

Find the indicated derivative. \(\frac{d z}{d x}\) if \(z=x^{2} \ln x^{2}+(\ln x)^{3}\)

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

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

The rate of change of velocity with respect to time is called acceleration. Suppose that the velocity at time \(t\) of a particle is given by \(v(t)=2 t^{2}\). Find the instantaneous acceleration when \(t=1\) second.

In Problems 21-28, find the indicated derivative. $$ y^{\prime} \text { where } y=\left(x^{2}+4\right)^{2} $$

Use the trigonometric identity \(\cos 2 x=2 \cos ^{2} x-1\) along with the Product Rule to find \(D_{x} \cos 2 x\).

Suppose that \(g(t)=a t^{2}+b t+c\) and \(g(1)=5\), \(g^{\prime}(1)=3\), and \(g^{\prime \prime}(1)=-4\). Find \(a, b\), and \(c\).

Find the indicated derivative. \(\frac{d r}{d x}\) if \(r=\frac{\ln x}{x^{2} \ln x^{2}}+\left(\ln \frac{1}{x}\right)^{3}\)

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

\(H(x)=\sqrt{x^{2}+4}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{2}{3 x}-\frac{2}{3} $$

A city is hit by an Asian flu epidemic. Officials estimate that \(t\) days after the beginning of the epidemic the number of persons sick with the flu is given by \(p(t)=120 t^{2}-2 t^{3}\), when \(0 \leq t \leq 40\). At what rate is the flu spreading at time \(t=10 ; t=20 ; t=40\) ?

In Problems 21-28, find the indicated derivative. $$ y^{\prime} \text { where } y=(x+\sin x)^{2} $$

Find the indicated derivative. \(g^{\prime}(x)\) if \(g(x)=\ln \left(x+\sqrt{x^{2}+1}\right)\)

Find \(D_{x} y\). $$ y=\tanh (\cot x) $$

In Problems 23-26, use \(f^{\prime}(x)=\lim _{t \rightarrow x}[f(t)-f(x)] /[t-x]\) to find \(f^{\prime}(x)\) (see Example 5). 23\. \(f(x)=x^{2}-3 x\)

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

In Problems 21-28, find the indicated derivative. $$ D_{t}\left(\frac{3 t-2}{t+5}\right)^{3} $$

A Ferris wheel of radius 20 feet is rotating counterclockwise with an angular velocity of 1 radian per second. One seat on the rim is at \((20,0)\) at time \(t=0\). (a) What are its coordinates at \(t=\pi / 6\) ? (b) How fast is it rising (vertically) at \(t=\pi / 6\) ? (c) How fast is it rising when it is rising at the fastest rate? 25\. Find the equation of the tangent line to \(y=\tan x\) at \(x=0\).

In Problems 23-28, an object is moving along a horizontal coordinate line according to the formula \(s=f(t)\), where \(s\), the directed distance from the origin, is in feet and \(t\) is in seconds. In each case, answer the following questions (see Examples 2 and 3). (a) What are \(v(t)\) and \(a(t)\), the velocity and acceleration, at time \(t\) ? (b) When is the object moving to the right? (c) When is it moving to the left? (d) When is its acceleration negative? (e) Draw a schematic diagram that shows the motion of the object. $$ s=t^{3}-6 t^{2} $$

Find the indicated derivative. \(h^{\prime}(x)\) if \(h(x)=\ln \left(x+\sqrt{x^{2}-1}\right)\)

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

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

In Problems 21-28, find the indicated derivative. $$ D_{s}\left(\frac{s^{2}-9}{s+4}\right) $$

Find the equation of the tangent line to \(y=\tan x\) at \(x=0\).

In Problems 23-28, an object is moving along a horizontal coordinate line according to the formula \(s=f(t)\), where \(s\), the directed distance from the origin, is in feet and \(t\) is in seconds. In each case, answer the following questions (see Examples 2 and 3). (a) What are \(v(t)\) and \(a(t)\), the velocity and acceleration, at time \(t\) ? (b) When is the object moving to the right? (c) When is it moving to the left? (d) When is its acceleration negative? (e) Draw a schematic diagram that shows the motion of the object. $$ s=t^{3}-9 t^{2}+24 t $$

$$ y=\frac{1}{\left(x^{3}+2 x\right)^{2 / 3}} $$

Find the indicated derivative. \(f^{\prime}(81)\) if \(f(x)=\ln \sqrt[3]{x}\)

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

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

In Problems 21-28, find the indicated derivative. $$ \frac{d}{d t}\left(\frac{(3 t-2)^{3}}{t+5}\right) $$

Find all points on the graph of \(y=\tan ^{2} x\) where the tangent line is horizontal.

In Problems 23-28, an object is moving along a horizontal coordinate line according to the formula \(s=f(t)\), where \(s\), the directed distance from the origin, is in feet and \(t\) is in seconds. In each case, answer the following questions (see Examples 2 and 3). (a) What are \(v(t)\) and \(a(t)\), the velocity and acceleration, at time \(t\) ? (b) When is the object moving to the right? (c) When is it moving to the left? (d) When is its acceleration negative? (e) Draw a schematic diagram that shows the motion of the object. $$ s=2 t^{3}-6 t+5 $$

Find the indicated derivative. \(f^{\prime}\left(\frac{\pi}{4}\right)\) if \(f(x)=\ln (\cos x)\)

Find \(D_{x} y\). $$ y=\arccos \left(e^{x}\right) $$

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=(-3 x+2)^{2} $$

In Problems 21-28, find the indicated derivative. $$ \frac{d}{d \theta}\left(\sin ^{3} \theta\right) $$

Find all points on the graph of \(y=9 \sin x \cos x\) where the tangent line is horizontal.

In Problems 23-28, an object is moving along a horizontal coordinate line according to the formula \(s=f(t)\), where \(s\), the directed distance from the origin, is in feet and \(t\) is in seconds. In each case, answer the following questions (see Examples 2 and 3). (a) What are \(v(t)\) and \(a(t)\), the velocity and acceleration, at time \(t\) ? (b) When is the object moving to the right? (c) When is it moving to the left? (d) When is its acceleration negative? (e) Draw a schematic diagram that shows the motion of the object. $$ s=t^{2}+\frac{16}{t}, t>0 $$

Find the indicated derivative. \(D_{x} e^{x+2}\)

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

\(\lim _{h \rightarrow 0} \frac{2(5+h)^{3}-2(5)^{3}}{h}\)

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\left(x^{2}+2\right)\left(x^{3}+1\right) $$

In Problems 21-28, find the indicated derivative. $$ \frac{d y}{d x}, \text { where } y=\left(\frac{\sin x}{\cos 2 x}\right)^{3} $$

An 18 -foot ladder leans against a 12 -foot vertical wall, its top extending over the wall. The bottom end of the ladder is pulled along the ground away from the wall at 2 feet per second. (a) Find the vertical velocity of the top end when the ladder makes an angle of \(60^{\circ}\) with the ground. (b) Find the vertical acceleration at the same instant.

Let \(f(x)=x-\sin x\). Find all points on the graph of \(y=f(x)\) where the tangent line is horizontal. Find all points on the graph of \(y=f(x)\) where the tangent line has slope 2 .

Find the indicated derivative. \(D_{x} e^{2 x^{2}-x}\)

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