91Ó°ÊÓ

Graphical Reasoning Consider the function \(f(x)=\frac{1}{3} x^{3}\) (a) Use a graphing utility to graph the function and estimate the values of \(f^{\prime}(0), f^{\prime}\left(\frac{1}{2}\right), f^{\prime}(1), f^{\prime}(2),\) and \(f^{\prime}(3) .\) (b) Use your results from part (a) to determine the values of \(\quad f^{\prime}\left(-\frac{1}{2}\right), f^{\prime}(-1), f^{\prime}(-2),\) and \(f^{\prime}(-3) .\) (c) Sketch a possible graph of \(f^{\prime}\) (d) Use the definition of derivative to find \(f^{\prime}(x)\)

Evaluating a Derivative In Exercises \(59-62\) , evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. $$\text {Function} \quad \text{Point}$$ $$ f(x)=\tan x \cot x \quad (1,1) $$

In Exercises 57–62, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. $$ y=x^{2}+9 $$

Finding a Derivative In Exercises \(43-64\) , find the derivative of the function. $$ y=3 x-5 \cos (\pi x)^{2} $$

Graphical Reasoning In Exercises 61 and \(62,\) use a graphing utility to graph the functions \(f\) and \(g\) in the same viewing window, where $$ g(x)=\frac{f(x+0.01)-f(x)}{0.01} $$ Label the graphs and describe the relationship between them. $$ f(x)=2 x-x^{2} $$

Evaluating a Derivative In Exercises \(59-62\) , evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. $$\text {Function} \quad \text{Point}$$ $$ h(t)=\frac{\sec t}{t} \quad \left(\pi,-\frac{1}{\pi}\right) $$

Use a graphing utility to sketch the intersecting graphs of the equations and show that they are orthogonal. [Two graphs are orthogonal if at their point(s) of intersection, their tangent lines are perpendicular to each other.] \(x+y=0\) \(x=\sin y\)

In Exercises 57–62, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line. $$ y=x+\sin x, \quad 0 \leq x<2 \pi $$

Finding a Derivative In Exercises \(43-64\) , find the derivative of the function. $$ y=\sqrt{x}+\frac{1}{4} \sin (2 x)^{2} $$

Finding a Derivative In Exercises \(43-64\) , find the derivative of the function. $$ y=\sin \sqrt[3]{x}+\sqrt[3]{\sin x} $$