Chapter 2

Given that $$ f^{\prime}(0)=2, g(0)=0 \quad \text { and } \quad g^{\prime}(0)=3 $$ find \((f \circ g)^{\prime}(0)\)

Find \(f^{\prime}(x)\) $$ f(x)=4 \cos x+2 \sin x $$

Find \(d y / d x\) $$ y=4 x^{7} $$

Compute the derivative of the given function \(f(x)\) by \((a)\) multiplying and then differentiating and (b) using the product rule. Verify that (a) and (b) yield the same result. \(f(x)=(x+1)(2 x-1)\)

Given that $$ f^{\prime}(9)=5, g(2)=9 \quad \text { and } \quad g^{\prime}(2)=-3 $$ find \((f \circ g)^{\prime}(2)\)

Find \(f^{\prime}(x)\) $$ f(x)=\frac{5}{x^{2}}+\sin x $$

Find \(d y / d x\) $$ y=-3 x^{12} $$

Compute the derivative of the given function \(f(x)\) by \((a)\) multiplying and then differentiating and (b) using the product rule. Verify that (a) and (b) yield the same result. \(f(x)=\left(3 x^{2}-1\right)\left(x^{2}+2\right)\)

Let \(f(x)=x^{5}\) and \(g(x)=2 x-3\) (a) Find \((f \circ g)(x)\) and \((f \circ g)^{\prime}(x)\) (b) Find \((g \circ f)(x)\) and \((g \circ f)^{\prime}(x)\)

Find \(f^{\prime}(x)\) $$ f(x)=-4 x^{2} \cos x $$

Find \(d y / d x\) $$ y=3 x^{8}+2 x+1 $$

Compute the derivative of the given function \(f(x)\) by \((a)\) multiplying and then differentiating and (b) using the product rule. Verify that (a) and (b) yield the same result. \(f(x)=\left(x^{2}+1\right)\left(x^{2}-1\right)\)

(a) If you are given an equation for the tangent line at the point \((a, f(a))\) on a curve \(y=f(x),\) how would you go about finding \(f^{\prime}(a) ?\) (b) Given that the tangent line to the graph of \(y=f(x)\) at the point \((2,5)\) has the equation \(y=3 x-1,\) find \(f^{\prime}(2) .\) (c) For the function \(y=f(x)\) in part (b), what is the instantaneous rate of change of \(y\) with respect to \(x\) at \(x=2 ?\)

Let \(f(x)=5 \sqrt{x}\) and \(g(x)=4+\cos x\) (a) Find \((f \circ g)(x)\) and \((f \circ g)^{\prime}(x)\) (b) Find \((g \circ f)(x)\) and \((g \circ f)^{\prime}(x)\)

Find \(f^{\prime}(x)\) $$ f(x)=2 \sin ^{2} x $$

Find \(d y / d x\) $$ y=\frac{1}{2}\left(x^{4}+7\right) $$

Compute the derivative of the given function \(f(x)\) by \((a)\) multiplying and then differentiating and (b) using the product rule. Verify that (a) and (b) yield the same result. \(f(x)=(x+1)\left(x^{2}-x+1\right)\)

Given that the tangent line to \(y=f(x)\) at the point \((1,2)\) passes through the point \((-1,-1),\) find \(f^{\prime}(1) .\)

Given the following table of values, find the indicated derivatives in parts (a) and (b). $$ \begin{array}{|c|c|c|c|c|}\hline x & {f(x)} & {f^{\prime}(x)} & {g(x)} & {g^{\prime}(x)} \\ \hline 3 & {5} & {-2} & {5} & {7} \\ \hline 5 & {3} & {-1} & {12} & {4} \\ \hline\end{array} $$ (a) \(F^{\prime}(3),\) where \(F(x)=f(g(x))\) (b) \(G^{\prime}(3),\) where \(G(x)=g(f(x))\)

Find \(f^{\prime}(x)\) $$ f(x)=\frac{5-\cos x}{5+\sin x} $$

Find \(d y / d x\) $$ y=\pi^{3} $$

Find \(f^{\prime}(x)\). \(f(x)=\left(3 x^{2}+6\right)\left(2 x-\frac{1}{4}\right)\)

Sketch the graph of a function \(f\) for which \(f(0)=-1\) \(f^{\prime}(0)=0, f^{\prime}(x)<0\) if \(x<0,\) and \(f^{\prime}(x)>0\) if \(x>0\)

If a particle moves at constant velocity, what can you say about its position versus time curve?

Given the following table of values, find the indicated derivatives in parts (a) and (b). $$ \begin{array}{|c|c|c|c|c|}\hline x & {f(x)} & {f^{\prime}(x)} & {g(x)} & {g^{\prime}(x)} \\ \hline-1 & {2} & {3} & {2} & {-3} \\ \hline 2 & {0} & {4} & {1} & {-5} \\ \hline\end{array} $$ (a) \(F^{\prime}(-1),\) where \(F(x)=f(g(x))\) (b) \(G^{\prime}(-1),\) where \(G(x)=g(f(x))\)

Find \(f^{\prime}(x)\) $$ f(x)=\frac{\sin x}{x^{2}+\sin x} $$

Find \(d y / d x\) $$ y=\sqrt{2} x+(1 / \sqrt{2}) $$

Find \(f^{\prime}(x)\). \(f(x)=\left(2-x-3 x^{3}\right)\left(7+x^{5}\right)\)

Sketch the graph of a function \(f\) for which \(f(0)=0,\) \(f^{\prime}(0)=0,\) and \(f^{\prime}(x) > 0\) if \(x< 0\) or \(x > 0\)

An automobile, initially at rest, begins to move along a straight track. The velocity increases steadily until suddenly the driver sees a concrete barrier in the road and applies the brakes sharply at time \(t_{0}\). The car decelerates the brakes sharplate-the car crashes into the barrier at time \(t_{1}\) and instantaneously comes to rest. Sketch a position versus time curve that might represent the motion of the car. Indicate how characteristics of your curve correspond to the events of this scenario.

Find \(f^{\prime}(x)\) $$ f(x)=\left(x^{3}+2 x\right)^{37} $$

Find \(f^{\prime}(x)\) $$ f(x)=\sec x-\sqrt{2} \tan x $$

Find \(d y / d x\) $$ y=-\frac{1}{3}\left(x^{7}+2 x-9\right) $$

Find \(f^{\prime}(x)\). \(f(x)=\left(x^{3}+7 x^{2}-8\right)\left(2 x^{-3}+x^{-4}\right)\)

Given that \(f(3)=-1\) and \(f^{\prime}(3)=5,\) find an equation for the tangent line to the graph of \(y=f(x)\) at \(x=3\)

For each exercise, sketch a curve and a line \(L\) satisfying the stated conditions. $$ \begin{array}{l}{L \text { is tangent to the curve and intersects the curve in at }} \\ {\text { least two points. }}\end{array} $$

Find \(f^{\prime}(x)\) $$ f(x)=\left(3 x^{2}+2 x-1\right)^{6} $$

Find \(f^{\prime}(x)\) $$ f(x)=\left(x^{2}+1\right) \sec x $$

Find \(d y / d x\) $$ y=\frac{x^{2}+1}{5} $$

Find \(f^{\prime}(x)\). \(f(x)=\left(\frac{1}{x}+\frac{1}{x^{2}}\right)\left(3 x^{3}+27\right)\)

Given that \(f(-2)=3\) and \(f^{\prime}(-2)=-4,\) find an equation for the tangent line to the graph of \(y=f(x)\) at \(x=-2\)

Find \(f^{\prime}(x)\) $$ f(x)=\left(x^{3}-\frac{7}{x}\right)^{-2} $$

Find \(f^{\prime}(x)\) $$ f(x)=4 \csc x-\cot x $$

Find \(f^{\prime}(x)\) $$ f(x)=x^{-3}+\frac{1}{x^{7}} $$

Find \(f^{\prime}(x)\). \(f(x)=(x-2)\left(x^{2}+2 x+4\right)\)

Find \(f^{\prime}(x)\) $$ f(x)=\frac{1}{\left(x^{5}-x+1\right)^{9}} $$

Find \(f^{\prime}(x)\) $$ f(x)=\cos x-x \csc x $$

Find \(f^{\prime}(x)\). \(f(x)=\left(x^{2}+x\right)\left(x^{2}-x\right)\)

Find \(f^{\prime}(x)\) $$ f(x)=\sqrt{x}+\frac{1}{x} $$

Find \(f^{\prime}(x)\) $$ f(x)=\frac{4}{\left(3 x^{2}-2 x+1\right)^{3}} $$