91Ó°ÊÓ

Assume that \(x\) and \(y\) are both differentiable functions of \(t\) and find the required values of \(d y / d t\) and \(d x / d t\) $$ \begin{aligned} &x^{2}+y^{2}=25 \quad \text { (a) } \frac{d y}{d t} \text { when } x=3, y=4 \quad \frac{d x}{d t}=8\\\ &\text { (b) } \frac{d x}{d t} \text { when } x=4, y=3 \quad \frac{d y}{d t}=-2 \end{aligned} $$

Find the derivative of the function. $$ f(x)=\sqrt[5]{x} $$

In Exercises \(5-10\), find the slope of the tangent line to the graph of the function at the given point. \(f(x)=3-2 x, \quad(-1,5)\)

In Exercises 5-8, show that the slopes of the graphs of \(f\) and \(f^{-1}\) are reciprocals at the indicated points. $$\begin{array}{ll}\text { Function } & \text { Point } \\\\\hline f(x)=x^{3} & \left(\frac{1}{2}, \frac{1}{8}\right)\\\f^{-1}(x)=\sqrt[3]{x} & \left(\frac{1}{8}, \frac{1}{2}\right)\end{array}$$

Use the Product Rule to differentiate the function. $$ f(x)=x^{3} \cos x $$

A point is moving along the graph of the given function such that \(d x / d t\) is 2 centimeters per second. Find \(d y / d t\) for the given values of \(x\). $$ y=x^{2}+1 \quad \text { (a) } x=-1 \quad \text { (b) } x=0 \quad \text { (c) } x=1 $$

In Exercises \(5-12,\) approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than \(0.001 .\) Then find the zero(s) using a graphing utility and compare the results. $$ f(x)=x^{3}+x-1 $$

Find \(d y / d x\) by implicit differentiation. $$ x^{3}-x y+y^{2}=4 $$

A point is moving along the graph of the given function such that \(d x / d t\) is 2 centimeters per second. Find \(d y / d t\) for the given values of \(x\). $$ y=\frac{1}{1+x^{2}} \quad \text { (a) } x=-2 \quad \text { (b) } x=0 \quad \text { (c) } x=2 $$

In Exercises \(5-12,\) approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than \(0.001 .\) Then find the zero(s) using a graphing utility and compare the results. $$ f(x)=x-2 \sqrt{x+1} $$