91Ó°ÊÓ

Differentiate two ways: first, by using the Quotient Rule; then, by dividing the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(g(x)=\frac{3 x^{7}-x^{3}}{x}\)

(a) find the simplified form of the difference quotient and then (b) complete the following table. $$ \begin{array}{|c|l|l|} \hline x & h & \frac{f(x+h)-f(x)}{h} \\ \hline 5 & 2 & \\ \hline 5 & 1 & \\ \hline 5 & 0.1 & \\ \hline 5 & 0.01 & \\ \hline \end{array} $$ $$ f(x)=x^{2}-3 x+5 $$

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 2}\left(2 x^{4}-3 x^{3}+4 x-1\right) $$

Find \(f^{\prime \prime}(x)\) $$ f(x)=x^{1 / 5} $$

Find \(\frac{d y}{d x}\). $$ y=4 \sqrt{x} $$

Differentiate each function $$ y=\frac{4 x^{2}}{(7-5 x)^{3}} $$

a) Graph the function. b) Draw tangent lines to the graph at points whose \(x\) -coordinates are \(-2,0,\) and 1 c) Find \(f^{\prime}(x)\) by determining \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\). d) Find \(f^{\prime}(-2), f^{\prime}(0),\) and \(f^{\prime}(1) .\) These slopes should match those of the lines you drew in part (b). $$f(x)=\frac{2}{x}$$

Find \(f^{\prime \prime}(x)\) $$ f(x)=x^{1 / 3} $$

Differentiate two ways: first, by using the Quotient Rule; then, by dividing the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(f(x)=\frac{2 x^{5}+x^{2}}{x}\)

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow-1}\left(3 x^{5}+4 x^{4}-3 x+6\right) $$