91Ó°ÊÓ

Find the derivative of each function in two ways: a. Using the Product Rule. b. Multiplying out the function and using the Power Rule. Your answers to parts (a) and (b) should agree. $$ x^{4}\left(x^{5}+1\right) $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \left(x^{2}-x\right)^{-3} $$

Complete the tables and use them to find the given limits. Round calculations to three decimal places. A graphing calculator with a TABLE feature will be very helpful. a. \(\lim _{x \rightarrow 1^{-}}\left(\frac{x^{4}-1}{x-1}\right)\) b. \(\lim _{x \rightarrow 1^{-}}\left(\frac{x^{4}-1}{x-1}\right)\) c. \(\lim _{x \rightarrow 1}\left(\frac{x^{4}-1}{x-1}\right)\)

Find the derivative of each function in two ways: a. Using the Product Rule. b. Multiplying out the function and using the Power Rule. Your answers to parts (a) and (b) should agree. $$ x^{5}\left(x^{4}+1\right) $$

Find the derivative of each function. $$ f(x)=x^{1000} $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \frac{1}{x^{2}+x} $$

Find the derivative of each function by using the Product Rule. Simplify your answers. $$ f(x)=x^{2}\left(x^{3}+1\right) $$

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$ \frac{x^{3}+1}{x^{3}-1} $$

Use the definition of the derivative to show that the following functions are not differentiable at \(x=0\). \(f(x)=|2 x|\)

Find the derivative of each function. $$ f(x)=x^{1 / 2} $$