91Ó°ÊÓ

Finding a Derivative In Exercises \(37-58\) , find the derivative of the function. (Hint: In some exercises, you may find it helpful to apply logarithmic properties before differentiating.) $$ y=\log _{5} \sqrt{x^{2}-1} $$

Find the derivative of the function. \(y=\frac{1}{2}\left(\frac{1}{2} \ln \frac{x+1}{x-1}+\arctan x\right)\)

Finding an Integral Decide whether you can find the integral $$\int \frac{2 d x}{\sqrt{x^{2}+4}}$$ using the formulas and techniques you have studied so far. Explain your reasoning.

Finding a Derivative In Exercises \(37-58\) , find the derivative of the function. (Hint: In some exercises, you may find it helpful to apply logarithmic properties before differentiating.) $$ f(x)=\log _{2} \sqrt[3]{2 x+1} $$

In Exercises 43–54, find the indefinite integral. $$ \int \operatorname{sech}^{3} x \tanh x d x $$

Find the derivative of the function. \(y=\frac{1}{2}\left[x \sqrt{4-x^{2}}+4 \arcsin \left(\frac{x}{2}\right)\right]\)

Determine whether the function is one-to-one. If it is, find its inverse function. \(f(x)=-3\)

Finding a Derivative In Exercises \(33-54,\) find the derivative. $$ y=e^{2 x} \tan 2 x $$

Evaluating a Definite Integral In Exercises \(49-56\) , evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{e}^{e^{2}} \frac{1}{x \ln x} d x $$

In Exercises 41–64, find the derivative of the function. $$ f(x)=\ln \left(\frac{2 x}{x+3}\right) $$