91Ó°ÊÓ

A point is moving along the graph of \(y=1 /\left(1+x^{2}\right)\) such that \(d x / d t\) is 2 centimeters per minute. Find \(d y / d t\) for each value of \(x\). (a) \(x=-2\) (b) \(x=2\) (c) \(x=0\) (d) \(x=10\)

In Exercises, find the second derivative of the function. $$ h(s)=s^{3}\left(s^{2}-2 s+1\right) $$

In Exercises, find \(d y / d x\) by implicit differentiation and evaluate the derivative at the given point. $$ x^{3}-y^{2}=0 $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ y=(x-2)^{3} $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\sqrt{2 x^{2}+6} $$

In Exercises, use a graphing utility to graph the function. Then find all relative extrema of the function. $$ f(x)=x+\frac{1}{x} $$

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=\sqrt{x^{2}-1} $$

In Exercises, find all relative extrema of the function. Use the Second- Derivative Test when applicable. $$ f(x)=\sqrt{9-x^{2}} $$

In Exercises, find the third derivative of the function. $$ f(x)=x^{5}-3 x^{4} $$

In Exercises, use a graphing utility to graph the function. Then find all relative extrema of the function. $$ f(x)=\frac{x}{x+1} $$