91Ó°ÊÓ

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln (\sec (\ln \theta)) $$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\ln \left(\tan ^{-1} x\right) $$

Endangered species Biologists consider a species of animal or plant to be endangered if it is expected to become extinct within 20 years. If a certain species of wildlife is counted to have 1147 members at the present time, and the population has been steadily declining exponentially at an annual rate averaging 39\(\%\) over the past 7 years, do you think the species is endangered? Explain your answer.

Evaluate the integrals. \begin{equation}\int\left(e^{3 x}+5 e^{-x}\right) d x\end{equation}

Evaluate the integrals. \begin{equation}\int\left(2 e^{x}-3 e^{-2 x}\right) d x\end{equation}

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\tan ^{-1}(\ln x) $$

In Exercises \(25-36,\) find the derivative of \(y\) with respect to the appropriate variable. $$y=\operatorname{csch}^{-1} 2^{\theta}$$

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln \left(\frac{\sqrt{\sin \theta \cos \theta}}{1+2 \ln \theta}\right) $$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{x \rightarrow 0^{+}} \frac{\ln \left(e^{x}-1\right)}{\ln x} $$

Each of Exercises \(25-34\) gives a formula for a function \(y=f(x)\) . In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1}\) . As a check, show that \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.\) $$f(x)=\left(2 x^{3}+1\right)^{1 / 5}$$