91Ó°ÊÓ

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 \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.\) $$f(x)=x^{3}+1$$

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

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{\theta \rightarrow 0} \frac{3^{\sin \theta}-1}{\theta} $$

In Exercises \(21-42,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=\csc ^{-1}\left(x^{2}+1\right), x>0 $$

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

find \(d y / d x.\) \begin{equation}\ln y=e^{y} \sin x\end{equation}

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 \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.\) $$f(x)=(1 / 2) x-7 / 2$$

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

Use l'Hopital's rule to find the limits in Exercises \(7-50\) . $$ \lim _{\theta \rightarrow 0} \frac{(1 / 2)^{\theta}-1}{\theta} $$

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