91Ó°ÊÓ

Differentiate the function. $$ y=\sqrt{1+x e^{-2 x}} $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow \infty}(x-\ln x)$$

(a) Show that \(f\) is one-to-one. (b) Use Theorem 7 to find \(\left(f^{-1}\right)^{\prime}(a)\) . (c) Calculate \(f^{-1}(x)\) and state the domain and range of \(f^{-1}\) (d) Calculate \(f^{-1}(x)\) and state the formula in part (c) and check that it agrees with the result of part (b). (e) Sketch the graphs of \(f\) and \(f^{-1}\) on the same axes. $$ f(x)=x^{3}, \quad a=8 $$

Find the derivative of the function. Find the domains of the function and its derivative. \(g(x)=\cos ^{-1}(3-2 x)\)

Differentiate the function. $$ f(t)=\tan \left(e^{t}\right)+e^{\tan t} $$

If you graph the function \(f(x)=\frac{1-e^{1 / x}}{1+e^{1 / x}}\) you'll see that \(f\) appears to be an odd function. Prove it.

$$y^{\prime} \text { if } \tan ^{-1}(x y)=1+x^{2} y$$

Differentiate the function. $$ y=e^{k \tan \sqrt{x}} $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 1^{+}}\left[\ln \left(x^{7}-1\right)-\ln \left(x^{5}-1\right)\right]$$

Find \(y^{\prime}\) if tan \(^{-1}(x y)=1+x^{2} y.\)