Chapter 9: Chapter 9

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=9 x^{1 / 3}\)

Complete the table by computing \(f(x)\) at the given values of \(x\). Use these results to estimate the indicated limit (if it exists). $$ \begin{array}{l} f(x)=\frac{|x|}{x} ; \lim _{x \rightarrow 0} f(x)\\\ \begin{array}{lllllll} \hline \boldsymbol{x} & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} \end{array} $$

Find the derivative of each function. \(f(x)=\left(5 x^{2}+1\right)(2 \sqrt{x}-1)\)

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

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

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

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

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

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. \(f(x)=8-4 x\)

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