Chapter 2

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

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

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

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

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

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

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

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

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

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

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 \(f^{\prime}(1)\)

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). $$\begin{array}{|c|c|c|c|c|} \hline x & f(x) & f^{\prime}(x) & g(x) & g^{\prime}(x) \\ \hline 3 & 5 & -2 & 5 & 7 \\ \hline 5 & 3 & -1 & 12 & 4 \\\ \hline \end{array}$$ (a) \(F^{\prime}(3),\) where \(F(x)=f(g(x))\) (b) \(G^{\prime}(3),\) where \(G(x)=g(f(x))\)

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

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 \(x>0\)

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

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

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 rapidly, but it is too late - 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.

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

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 \(x>0\)

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

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

Sketch a curve and a line \(L\) satisfying the stated conditions. \(L\) is tangent to the curve and intersects the curve in at least two points.

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

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

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 \(x=3\)

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

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

Sketch a curve and a line \(L\) satisfying the stated conditions. \(L\) intersects the curve in exactly one point, but \(L\) is not tangent to the curve.

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

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

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

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

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

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

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

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

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

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

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

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