91Ó°ÊÓ

A function \(y=f(x)\) and values of \(x_{0}\) and \(x_{1}\) are given. (a) Find the average rate of change of \(y\) with respect to \(x\) over the interval \(\left[x_{0}, x_{1}\right]\). (b) Find the instantaneous rate of change of \(y\) with respect to \(x\) at the specified value of \(x_{0}\). (c) Find the instantaneous rate of change of \(y\) with respect to \(x\) at an arbitrary value of \(x_{0}\). (d) The average rate of change in part (a) is the slope of a certain secant line, and the instantaneous rate of change in part (b) is the slope of a certain tangent line. Sketch the graph of \(y=f(x)\) together with those two lines. $$ y=2 x^{2} ; x_{0}=0, x_{1}=1 $$

Find \(f^{\prime}(x)\). $$ f(x)=-3 x^{-8}+2 \sqrt{x} $$

Find \(f^{\prime}(x)\) $$ f(x)=\frac{3 x+4}{x^{2}+1} $$

Find \(f^{\prime}(x)\). $$ f(x)=\csc x \cot x $$

A function \(y=f(x)\) and values of \(x_{0}\) and \(x_{1}\) are given. (a) Find the average rate of change of \(y\) with respect to \(x\) over the interval \(\left[x_{0}, x_{1}\right]\). (b) Find the instantaneous rate of change of \(y\) with respect to \(x\) at the specified value of \(x_{0}\). (c) Find the instantaneous rate of change of \(y\) with respect to \(x\) at an arbitrary value of \(x_{0}\). (d) The average rate of change in part (a) is the slope of a certain secant line, and the instantaneous rate of change in part (b) is the slope of a certain tangent line. Sketch the graph of \(y=f(x)\) together with those two lines. $$ y=x^{3} ; x_{0}=1, x_{1}=2 $$

Find \(f^{\prime}(x)\). $$ f(x)=7 x^{-6}-5 \sqrt{x} $$

Find \(f^{\prime}(x)\). \(f(x)=\sqrt{x^{3}-2 x+5}\)

Find \(f^{\prime}(x)\) $$ f(x)=\frac{x-2}{x^{4}+x+1} $$

Find \(f^{\prime}(x)\). $$ f(x)=\frac{\cot x}{1+\csc x} $$

Find \(f^{\prime}(x)\). $$ f(x)=x^{e}+\frac{1}{x^{\sqrt{10}}} $$