91Ó°ÊÓ

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\ln \frac{3}{x}$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) $$\lim _{t \rightarrow-3} \frac{t^{3}-4 t+15}{t^{2}-t-12}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=x e^{x}-e^{x}$$

True, or false? As \(x \rightarrow \infty,\) \begin{equation} \begin{array}{ll}{\text { a. } x=o(x)} & {\text { b. } x=o(x+5)} \\ {\text { c. } x=O(x+5)} & {\text { d. } x=O(2 x)} \\ {\text { e. } e^{x}=o\left(e^{2 x}\right)} & {\text { f. } x+\ln x=O(x)} \\ {\text { g. } \ln x=o(\ln 2 x)} & {\text { h. } \sqrt{x^{2}+5}=O(x)} \end{array} \end{equation}

Solve the differential equations in Exercises \(9-22\) $$ 2 \sqrt{x y} \frac{d y}{d x}=1, \quad x, y>0 $$

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

Rewrite the expressions in Exercises \(5-10\) in terms of exponentials and simplify the results as much as you can. \(\ln (\cosh x+\sinh x)+\ln (\cosh x-\sinh x)\)

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=\ln \frac{10}{x}$$

Use l'Hopital's rule to find the limits in Exercises \(7-50\) $$\lim _{t \rightarrow-1} \frac{3 t^{3}+3}{4 t^{3}-t+3}$$

Solve the differential equations in Exercises \(9-22\) $$ \frac{d y}{d x}=x^{2} \sqrt{y}, \quad y > 0 $$