Chapter 9

Find the derivative of each function. \(f(x)=\left(x^{3}+2 x+1\right)\left(2+\frac{1}{x^{2}}\right)\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(u)=\frac{2}{\sqrt{u}}\)

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. \(f(x)=-\frac{1}{2} x^{2}\)

Complete the table by computing \(f(x)\) at the given values of \(x\). Use these results to estimate the indicated limit (if it exists). $$ \begin{array}{l} f(x)=\frac{1}{x-2} ; \lim _{x \rightarrow 2} f(x) \\ \hline \boldsymbol{x} \quad 1.9 \quad 1.99 \quad 1.999 \quad 2.001 \quad 2.01 \quad 2.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) \end{array} $$

Find the derivative of each function. \(f(x)=\frac{1}{(2 x+3)^{3}}\)

Find the derivative of each function. \(f(x)=\frac{1}{x-2}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=7 x^{-12}\)

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. \(f(x)=-x^{2}+3 x\)

Find the derivative of each function. \(f(x)=\frac{2}{\left(x^{2}-1\right)^{4}}\)

Find the derivative of each function. \(g(x)=\frac{3}{2 x+4}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=0.3 x^{-1.2}\)

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. \(f(x)=2 x^{2}+5 x\)

Complete the table by computing \(f(x)\) at the given values of \(x\). Use these results to estimate the indicated limit (if it exists). $$ \begin{array}{l} f(x)=\frac{x-1}{x-1} ; \lim _{x \rightarrow 1} f(x) \\ \hline \boldsymbol{x} \quad 0.9 \quad 0.99 \quad 0.999 \quad 1.001 \quad 1.01 \quad 1.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & \multicolumn{3}{|c} {} \\ \hline \end{array} $$

Find the derivative of each function. \(f(t)=\frac{1}{\sqrt{2 t-3}}\)

Find the derivative of each function. \(f(x)=\frac{x-1}{2 x+1}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=5 x^{2}-3 x+7\)

In Exercises 17-22, find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line. \(f(x)=2 x+7\) at \((2,11)\)

In Exercises 17-22, sketch the graph of the function \(f\) and evaluate \(\lim _{x \rightarrow a} f(x)\), if it exists, for the given value of \(a\). \(f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x \leq 0 \\ -1 & \text { if } x>0\end{array} \quad(a=0)\right.\)

Find the derivative of each function. \(f(x)=\frac{1}{\sqrt{2 x^{2}-1}}\)

Find the derivative of each function. \(f(t)=\frac{1-2 t}{1+3 t}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=x^{3}-3 x^{2}+1\)

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line. \(f(x)=-3 x+4\) at \((-1,7)\)

Sketch the graph of the function \(f\) and evaluate \(\lim _{x \rightarrow a} f(x)\), if it exists, for the given value of \(a\). \(f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x \leq 3 \\ -2 x+8 & \text { if } x>3\end{array} \quad(a=3)\right.\)

Find the derivative of each function. \(y=\frac{1}{\left(4 x^{4}+x\right)^{3 / 2}}\)

Find the derivative of each function. \(f(x)=\frac{1}{x^{2}+1}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=-x^{3}+2 x^{2}-6\)

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line. \(f(x)=3 x^{2}\) at \((1,3)\)

Sketch the graph of the function \(f\) and evaluate \(\lim _{x \rightarrow a} f(x)\), if it exists, for the given value of \(a\). \(f(x)=\left\\{\begin{array}{ll}x & \text { if } x<1 \\ 0 & \text { if } x=1 \\\ -x+2 & \text { if } x>1\end{array} \quad(a=1)\right.\)

Find the derivative of each function. \(f(t)=\frac{4}{\sqrt[3]{2 t^{2}+t}}\)

Find the derivative of each function. \(f(u)=\frac{u}{u^{2}+1}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=x^{4}-2 x^{2}+5\)

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line. \(f(x)=3 x-x^{2}\) at \((-2,-10)\)

Sketch the graph of the function \(f\) and evaluate \(\lim _{x \rightarrow a} f(x)\), if it exists, for the given value of \(a\). \(f(x)=\left\\{\begin{array}{ll}-2 x+4 & \text { if } x<1 \\ 4 & \text { if } x=1 \\ x^{2}+1 & \text { if } x>1\end{array} \quad(a=1)\right.\)

Find the derivative of each function. \(f(x)=\left(3 x^{2}+2 x+1\right)^{-2}\)

Find the derivative of each function. \(f(s)=\frac{s^{2}-4}{s+1}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=0.03 x^{2}-0.4 x+10\)

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line. \(f(x)=-\frac{1}{x}\) at \(\left(3,-\frac{1}{3}\right)\)

In Exercises 21-38, find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 1^{+}}(2 x+4)\)

Sketch the graph of the function \(f\) and evaluate \(\lim _{x \rightarrow a} f(x)\), if it exists, for the given value of \(a\). \(f(x)=\left\\{\begin{array}{ll}|x| & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{array} \quad(a=0)\right.\)

Find the derivative of each function. \(f(t)=\left(5 t^{3}+2 t^{2}-t+4\right)^{-3}\)

Find the derivative of each function. \(f(x)=\frac{x^{3}-2}{x^{2}+1}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=0.002 x^{3}-0.05 x^{2}+0.1 x-20\)

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line. \(f(x)=\frac{3}{2 x}\) at \(\left(1, \frac{3}{2}\right)\)

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 1^{-}}(3 x-4)\)

Sketch the graph of the function \(f\) and evaluate \(\lim _{x \rightarrow a} f(x)\), if it exists, for the given value of \(a\). \(f(x)=\left\\{\begin{array}{ll}|x-1| & \text { if } x \neq 1 \\ 0 & \text { if } x=1\end{array} \quad(a=1)\right.\)

Find the derivative of each function. \(f(x)=\left(x^{2}+1\right)^{3}-\left(x^{3}+1\right)^{2}\)

Find the derivative of each function. \(f(x)=\frac{\sqrt{x}+1}{x^{2}+1}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=\frac{x^{3}-4 x^{2}+3}{x}\)

Let \(f(x)=2 x^{2}+1\). a. Find the derivative \(f^{\prime}\) of \(f\). b. Find an equation of the tangent line to the curve at the point \((1,3)\). c. Sketch the graph of \(f\).

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 2^{-}} \frac{x-3}{x+2}\)