Chapter 7

Find the derivative of each of the following functions. $$y=6 x^{5}-x+10$$

Find the derivative of each of the following functions. $$f(x)=\frac{1}{x}+\frac{1}{\sqrt[3]{x^{2}}}$$

Find the derivative of each of the following functions. $$y=\frac{5 x^{6}-1}{x^{2}}$$

Find the derivative of each of the following functions. $$y=\frac{x^{2}}{5 x^{6}-1}$$

Find the derivative of each of the following functions. $$f(x)=(3 x-2)^{5}\left(x^{2}-1\right)$$

Find the derivative of each of the following functions. $$y=\sqrt{\frac{2 x+1}{2 x-1}}$$

Find the derivative of each of the following functions. $$y=10 \cot (2 x-1)$$

Find the derivative of each of the following functions. $$y=3 x \sec (3 x)$$

Find the derivative of each of the following functions. $$y=10 \cos \left[\sin \left(x^{2}-4\right)\right]$$

Find the derivative of each of the following functions. $$y=8 \cos ^{-1}(2 x)$$

Find the derivative of each of the following functions. $$y=3 e^{5}+4 x e^{x}$$

Find the derivative of each of the following functions. $$y=\ln \left(x^{2}+3\right)$$

Find \(\frac{d y}{d x},\) if \(x^{2}+y^{3}=10-5 x y\).

Let \(f\) be a continuous and differentiable function. Selected values of \(f\) are shown below. Find the approximate value of \(f^{\prime}\) at \(x=2\). $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 \\ \hline f & 6 & 5 & 6 & 9 & 14 \\ \hline \end{array} $$

If \(f(x)=x^{5}+3 x-8,\) find \(\left(f^{1}\right)^{\prime}(-8)\).

Write an equation of the tangent to the curve \(y=\ln x\) at \(x=e\).

If \(y=2 x \sin x,\) find \(\frac{d^{2} y}{d x^{2}}\) at \(x=\frac{\pi}{2}\).

If the function \(f(x)=(x-1)^{2 / 3}+2,\) find all points where \(f\) is not differentiable.

Write an equation of the normal line to the curve \(x \cos y=1\) at \(\left(2, \frac{\pi}{3}\right)\).

\(\lim _{x \rightarrow 3} \frac{x^{2}-3 x}{x^{2}-9}\)

\(\lim _{x \rightarrow 0^{+}} \frac{\ln (x+1)}{\sqrt{x}}\)

\(\lim _{x \rightarrow \infty} \frac{5 x+2 \ln x}{x+3 \ln x}\)