91Ó°ÊÓ

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=e^{\sin t}\left(\ln t^{2}+1\right)$$

Gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1} .\) Find a formula for \(f^{-1}\) in each case. $$f(x)=x^{2}-2 x+1, \quad x \geq 1$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) $$\lim _{x \rightarrow \pi / 2} \frac{\ln (\csc x)}{(x-(\pi / 2))^{2}}$$

Solve the differential equations in Exercises \(9-22\) . $$ \frac{d y}{d x}=e^{x-y}+e^{x}+e^{-y}+1 $$

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\frac{x \ln x}{1+\ln x}$$

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\ln (\ln x)$$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\sin ^{-1} \sqrt{2} t $$

\begin{equation} \begin{array}{l}{\text { a. Suppose you have three different algorithms for solving the }} \\ \quad {\text { same problem and each algorithm takes a number of steps that }} \\ \quad {\text { is of the order of one of the functions listed here: }}\end{array} \end{equation} $$n \log _{2} n, \quad n^{3 / 2}, \quad n\left(\log _{2} n\right)^{2}$$ \begin{equation} \begin{array}{l}{\text { Which of the algorithms is the most efficient in the long run? }} \\ {\text { Give reasons for your answer. }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { b. Graph the functions in part (a) together to get a sense of how }} \\ \quad {\text { rapidly each one grows. }}\end{array} \end{equation}

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. \(y=\left(x^{2}+1\right) \operatorname{sech}(\ln x)\) (Hint: Before differentiating, express in terms of exponentials and simplify.)

Gives a formula for a function \(y=f(x)\) and shows the graphs of \(f\) and \(f^{-1} .\) Find a formula for \(f^{-1}\) in each case. $$f(x)=(x+1)^{2}, \quad x \geq-1$$