Chapter 3

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ \frac{d}{d x} g(\tan 2 x) $$

A ball rolls down a long inclined plane so that its distance \(s\) from its starting point after \(t\) seconds is \(s=4.5 t^{2}+2 t\) feet. When will its instantaneous velocity be 30 feet per second?

First find and simplify $$\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ Then find \(d y / d x\) by taking the limit of your answer as \(\Delta x \rightarrow 0 .\) $$ y=\frac{x^{2}-1}{x} $$

, find dy/dx by logarithmic differentiation. Find and simplify \(f^{\prime}(1)\) if $$ f(x)=\ln \left(\frac{a x-b}{a x+b}\right)^{c}, \text { where } c=\frac{a^{2}-b^{2}}{2 a b} . $$

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ D_{x}\left(F(x) \sin ^{2} F(x)\right) $$

There are two tangent lines to the curve \(y=4 x-x^{2}\) that go through \((2,5)\). Find the equations of both of them. Hint: Let \(\left(x_{0}, y_{0}\right)\) be a point of tangency. Find two conditions that \(\left(x_{0}, y_{0}\right)\) must satisfy. See Figure 4 .

Convince yourself that \(f(x)=\left(x^{x}\right)^{x}\) and \(g(x)=x^{\left(x^{x}\right)}\) are not the same function. Then find \(f^{\prime}(x)\) and \(g^{\prime}(x) .\) Note: When mathematicians write \(x^{x^{x}}\), they mean \(x^{\left(x^{x}\right)}\).

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ D_{x} \sec ^{3} F(x) $$

Given that \(f(0)=1\) and \(f^{\prime}(0)=2\), find \(g^{\prime}(0)\) where \(g(x)=\cos f(x)\)

Given that \(F(0)=2\) and \(F^{\prime}(0)=-1\), find \(G^{\prime}(0)\) where \(G(x)=\frac{x}{1+\sec F(2 x)}\)

Let \(P(a, b)\) be a point on the first quadrant portion of the curve \(y=1 / x\) and let the tangent line at \(P\) intersect the \(x\) -axis at \(A\). Show that triangle \(A O P\) is isosceles and determine its area.

Given that \(f(1)=2, f^{\prime}(1)=-1, g(1)=0 \quad\) and \(g^{\prime}(1)=1\), find \(F^{\prime}(1)\) where \(F(x)=f(x) \cos g(x)\).

The radius of a spherical watermelon is growing at a constant rate of 2 centimeters per week. The thickness of the rind is always one-tenth of the radius. How fast is the volume of the rind growing at the end of the fifth week? Assume that the radius is initially 0 .

Find the equation of the tangent line to the graph of \(y=1+x \sin 3 x\) at \(\left(\frac{\pi}{3}, 1\right) .\) Where does this line cross the \(x\) -axis?

Find all points on the graph of \(y=\sin ^{2} x\) where the tangent line has slope \(1 .\)

Find the equation of the tangent line to \(y=\left(x^{2}+1\right)^{-2}\) at \(\left(1, \frac{1}{4}\right)\)

Where does the tangent line to \(y=(2 x+1)^{3}\) at \((0,1)\) cross the \(x\) -axis?

Suppose that \(f(x+y)=f(x) f(y)\) for all \(x\) and \(y\). Show that if \(f^{\prime}(0)\) exists then \(f^{\prime}(a)\) exists and \(f^{\prime}(a)=f(a) f^{\prime}(0)\).

Where does the tangent line to \(y=\left(x^{2}+1\right)^{-2}\) at \(\left(1, \frac{1}{4}\right)\) cross the \(x\) -axis?

Let \(f(x)=\left\\{\begin{array}{ll}m x+b & \text { if } x<2 \\ x^{2} & \text { if } x \geq 2\end{array}\right.\) Determine \(m\) and \(b\) so that \(f\) is differentiable everywhere.

The symmetric derivative \(f_{s}(x)\) is defined by $$f_{s}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h}$$ Show that if \(f^{\prime}(x)\) exists then \(f_{s}(x)\) exists, but that the converse is false.

A wheel centered at the origin and of radius 10 centimeters is rotating counterclockwise at a rate of 4 revolutions per second. A point \(P\) on the rim is at \((10,0)\) at \(t=0\). (a) What are the coordinates of \(P\) at time \(t\) seconds? (b) At what rate is \(P\) rising (or falling) at time \(t=1\) ?

Let \(f\) be differentiable and let \(f^{\prime}\left(x_{0}\right)=m .\) Find \(f^{\prime}\left(-x_{0}\right)\) if (a) \(f\) is an odd function. (b) \(f\) is an even function.

Prove that the derivative of an odd function is an even function and that the derivative of an even function is an odd function.

Use a CAS to do Problems 71 and \(72 .\) Draw the graphs of \(f(x)=x^{3}-4 x^{2}+3\) and its derivative \(f^{\prime}(x)\) on the interval \([-2,5]\) using the same axes. (a) Where on this interval is \(f^{\prime}(x)<0 ?\) (b) Where on this interval is \(f(x)\) decreasing? (c) Make a conjecture. Experiment with other intervals and other functions to support this conjecture.

Draw the graphs of \(f(x)=\cos x-\sin (x / 2)\) and its derivative \(f^{\prime}(x)\) on the interval \([0,9]\) using the same axes. (a) Where on this interval is \(f^{\prime}(x)>0\) ? (b) Where on this interval is \(f(x)\) increasing? (c) Make a conjecture. Experiment with other intervals and other functions to support this conjecture.

Show that \(D_{x}|x|=|x| / x, x \neq 0 .\) Hint: Write \(|x|=\sqrt{x^{2}}\) and use the Chain Rule with \(u=x^{2}\).

Let \(f(0)=1\) and \(f^{\prime}(0)=2\). Find the derivative of \(f(f(x)-1)\) at \(x=0\).

Let \(f(0)=0\) and \(f^{\prime}(0)=2\). Find the derivative of \(f(f(f(f(x))))\) at \(x=0\).

Suppose that \(f\) is a differentiable function. (a) Find \(\frac{d}{d x} f(f(x))\). (b) Find \(\frac{d}{d x} f(f(f(x)))\). (c) Let \(f^{[n]}\) denote the function defined as follows: \(f^{[1]}=f\) and \(f^{[n]}=f \circ f^{[n-1]}\) for \(n \geq 2 .\) Thus \(f^{[2]}=f \circ f, f^{[3]}=\) \(f \circ f \circ f\), etc. Based on your results from parts (a) and (b), make a conjecture regarding \(\frac{d}{d x} f^{[n]} .\) Prove your conjecture.

Give a second proof of the Quotient Rule. Write $$D_{x}\left(\frac{f(x)}{g(x)}\right)=D_{x}\left(f(x) \frac{1}{g(x)}\right)$$ and use the Product Rule and the Chain Rule.

Suppose that \(f\) is differentiable and that there are real numbers \(x_{1}\) and \(x_{2}\) such that \(f\left(x_{1}\right)=x_{2}\) and \(f\left(x_{2}\right)=x_{1}\). Let \(g(x)=f(f(f(f(x))))\). Show that \(g^{\prime}\left(x_{1}\right)=g^{\prime}\left(x_{2}\right)\).