91Ó°ÊÓ

Finding a Derivative In Exercises \(33-54,\) find the derivative. $$ y=x e^{4 x} $$

In Exercises 37–40, find any relative extrema of the function. Use a graphing utility to confirm your result. $$ h(x)=2 \tanh x-x $$

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.) $$ y=6^{3 x-4} $$

Find the derivative of the function. \(f(t)=\arcsin t^{2}\)

Completing the Square In Exercises \(33-42,\) find or evaluate the integral by completing the square. $$ \int \frac{1}{(x-1) \sqrt{x^{2}-2 x}} d x $$

Finding an Indefinite Integral of a Trigonometric Function In Exercises \(31-40\) , find the indefinite integral. $$ \int(\sec 2 x+\tan 2 x) d x $$

Find the inverse function of \(f,(\mathbf{b})\) graph \(f\) and \(f^{-1}\) on the same set of coordinate axes, ( \(\mathbf{c} )\) describe the relationship between the graphs, and ( \(\mathbf{d} )\) state the domain and range of \(f\) and \(f^{-1} .\) \(f(x)=x^{2}, \quad x \geq 0\)

In Exercises 37–40, find the limit. $$ \lim _{x \rightarrow 5^{+}} \ln \frac{x}{\sqrt{x-4}} $$

Differential Equation In Exercises \(41-44,\) solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. $$ \frac{d y}{d x}=\frac{3}{2-x}, \quad(1,0) $$

Completing the Square In Exercises \(33-42,\) find or evaluate the integral by completing the square. $$ \int \frac{x}{x^{4}+2 x^{2}+2} d x $$