91Ó°ÊÓ

In Exercises \(41-56,\) find the derivative of the function. $$f(t)=\operatorname{arccsc}\left(-t^{2}\right)$$

In Exercises \(39-42,\) find the limit. $$\lim _{x \rightarrow 5^{+}} \ln \frac{x}{\sqrt{x-4}}$$

Evaluating a Limit In Exercises \(15-42\) , evaluate the limit, using L'Hopital's Rule if necessary. $$\lim _{x \rightarrow 1^{+}} \frac{\int_{1}^{x} \cos \theta d \theta}{x-1}$$

Finding an Inverse Function In Exercises \(35-46\) , (a) find the inverse function of \(f,\) (b) graph \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs, and (d) state the domains and ranges of \(f\) and \(f^{-1}\) . $$f(x)=\sqrt{x^{2}-4}, \quad x \geq 2$$

In Exercises \(33-54,\) find the derivative of the function. $$g(t)=\left(e^{-t}+e^{t}\right)^{3}$$

Finding a Derivative In Exercises \(39-60,\) find the derivative of the function. $$f(x)=x 9^{x}$$

Finding a Derivative In Exercises \(43-66,\) find the derivative of the function. $$f(x)=\ln 3 x$$

In Exercises 43-46, use the specified substitution to find or evaluate the integral. $$\begin{array}{l}{\int \sqrt{e^{t}-3} d t} \\\ {u=\sqrt{e^{t}-3}}\end{array}$$

Finding an Inverse Function In Exercises \(35-46\) , (a) find the inverse function of \(f,\) (b) graph \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs, and (d) state the domains and ranges of \(f\) and \(f^{-1}\) . $$f(x)=\sqrt[3]{x}-1$$

Evaluating a limit In Exercises \(43-62\) , (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hopital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). $$\lim _{x \rightarrow \infty} x \ln x$$