Chapter 3

Find \(d y / d x\). $$y=x^{2} \cot x-\frac{1}{x^{2}}$$

Give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$s=\frac{25}{t+5}, \quad-4 \leq t \leq 0$$

Find the first and second derivatives. $$y=\frac{x^{3}}{3}+\frac{x^{2}}{2}+e^{-x}$$

Using the definition, calculate the derivatives of the functions in Exercises \(1-6 .\) Then find the values of the derivatives as specified. $$r(s)=\sqrt{2 s+1} ; \quad r^{\prime}(0), r^{\prime}(1), r^{\prime}(1 / 2)$$

If \(x^{2}+y^{2}=25\) and \(d x / d t=-2,\) then what is \(d y / d t\) when \(x=3\) and \(y=-4 ?\)

Find a linearization at a suitably chosen integer near \(a\) at which the given function and its derivative are easy to evaluate. $$f(x)=x^{2}+2 x, \quad a=0.1$$

Let \(f(x)=x^{3}-3 x^{2}-1, x \geq 2 .\) Find the value of \(d f^{-1} / d x\) at the point \(x=-1=f(3)\).

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$y=\tan u, \quad u=\pi x^{2}$$

Use implicit differentiation to find \(d y / d x\). $$y^{2}=\frac{x-1}{x+1}$$

Find \(d y / d x\). $$f(x)=\sin x \tan x$$

At time \(t\), the position of a body moving along the s-axis is \(s=t^{3}-6 t^{2}+9 t\) m. a. Find the body's acceleration each time the velocity is zero. b. Find the body's speed each time the acceleration is zero. c. Find the total distance traveled by the body from \(t=0\) to \(t=2\)

Find the first and second derivatives. $$w=3 z^{-2}-\frac{1}{z}$$

Find the indicated derivatives. $$\frac{d y}{d x}\( if \)y=2 x^{3}$$

If \(x^{2} y^{3}=4 / 27\) and \(d y / d t=1 / 2,\) then what is \(d x / d t\) when \(x=2 ?\)

Find a linearization at a suitably chosen integer near \(a\) at which the given function and its derivative are easy to evaluate. $$f(x)=x^{-1}, \quad a=0.9$$

Let \(f(x)=x^{2}-4 x-5, x>2 .\) Find the value of \(d f^{-1} / d x\) at the point \(x=0=f(5)\).

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(f^{\prime}(g(x)) g^{\prime}(x)\). $$y=-\sec u, \quad u=\frac{1}{x}+7 x$$

Use implicit differentiation to find \(d y / d x\). $$x^{3}=\frac{2 x-y}{x+3 y}$$

Find \(d y / d x\). $$g(x)=\frac{\cos x}{\sin ^{2} x}$$

At time \(t \geq 0,\) the velocity of a body moving along the horizontal \(s\) -axis is \(v=t^{2}-4 t+3\) a. Find the body's acceleration each time the velocity is zero. b. When is the body moving forward? Backward? c. When is the body's velocity increasing? Decreasing?

Find the first and second derivatives. $$s=-2 r^{-1}+\frac{4}{t^{2}}$$

Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together. $$y=\frac{1}{x^{2}}, \quad(-1,1)$$

Find the values. $$\sin \left(\cos ^{-1}\left(\frac{\sqrt{2}}{2}\right)\right)$$

If \(L=\sqrt{x^{2}+y^{2}}, d x / d t=-1,\) and \(d y / d t=3,\) find \(d L / d t\) when \(x=5\) and \(y=12\).

Find a linearization at a suitably chosen integer near \(a\) at which the given function and its derivative are easy to evaluate. $$f(x)=2 x^{2}+3 x-3, \quad a=-0.9$$

Suppose that the differentiable function \(y=f(x)\) has an inverse and that the graph of \(f\) passes through the point (2,4) and has a slope of \(1 / 3\) there. Find the value of \(d f^{-1} / d x\) at \(x=4\).

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=(2 x+1)^{5}$$

Use implicit differentiation to find \(d y / d x\). $$x=\sec y$$

Find \(d y / d x\). $$y=x e^{-x} \sec x$$

The equations for free fall at the surfaces of Mars and Jupiter ( \(s\) in meters, \(t\) in seconds) are \(s=1.86 t^{2}\) on Mars and \(s=11.44 t^{2}\) on Jupiter. How long does it take a rock falling from rest to reach a velocity of \(27.8 \mathrm{m} / \mathrm{sec}\) (about \(100 \mathrm{km} / \mathrm{h}\) ) on each planet?

Find the first and second derivatives. $$y=6 x^{2}-10 x-5 x^{-2}$$

Find the indicated derivatives. $$\frac{d s}{d t}\( if \)s=\frac{t}{2 t+1}$$

Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together. $$y=x^{3}, \quad(-2,-8)$$

Find the values. $$\sec \left(\cos ^{-1} \frac{1}{2}\right)$$

If \(r+s^{2}+v^{3}=12, d r / d t=4,\) and \(d s / d t=-3,\) find \(d v / d t\) when \(r=3\) and \(s=1\).

Find a linearization at a suitably chosen integer near \(a\) at which the given function and its derivative are easy to evaluate. $$f(x)=1+x, \quad a=8.1$$

Suppose that the differentiable function \(y=g(x)\) has an inverse and that the graph of \(g\) passes through the origin with slope 2 Find the slope of the graph of \(g^{-1}\) at the origin.

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=(4-3 x)^{9}$$

Use implicit differentiation to find \(d y / d x\). $$x y=\cot (x y)$$

Find \(d y / d x\). $$y=(\sin x+\cos x) \sec x$$

A rock thrown vertically upward from the surface of the moon at a velocity of \(24 \mathrm{m} / \mathrm{sec}\) (about \(86 \mathrm{km} / \mathrm{h})\) reaches a height of \(s=24 t-0.8 t^{2} \mathrm{m}\) in \(t\) sec. a. Find the rock's velocity and acceleration at time \(t .\) (The acceleration in this case is the acceleration of gravity on the moon.) b. How long does it take the rock to reach its highest point? c. How high does the rock go? d. How long does it take the rock to reach half its maximum height? e. How long is the rock aloft?

Find the first and second derivatives. $$y=4-2 x-x^{-3}$$

Find the indicated derivatives. $$\frac{d v}{d t}\( if \)v=t-\frac{1}{t}$$

Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together. $$y=\frac{1}{x^{3}}, \quad\left(-2,-\frac{1}{8}\right)$$

Find the values. $$\tan \left(\sin ^{-1}\left(-\frac{1}{2}\right)\right)$$

If the original 24 m edge length \(x\) of a cube decreases at the rate of \(5 \mathrm{m} / \mathrm{min},\) when \(x=3 \mathrm{m}\) at what rate does the cube's a. surface area change? b. volume change?

Find a linearization at a suitably chosen integer near \(a\) at which the given function and its derivative are easy to evaluate. $$f(x)=\sqrt[3]{x}, \quad a=8.5$$

In Exercises \(9-22,\) write the function in the form \(y=f(u)\) and \(u=g(x) .\) Then find \(d y / d x\) as a function of \(x\). $$y=\left(1-\frac{x}{7}\right)^{-7}$$

Use implicit differentiation to find \(d y / d x\). $$x+\tan (x y)=0$$

Find \(d y / d x\). $$y=\frac{\cot x}{1+\cot x}$$