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Find \(d^{2} y / d x^{2}\) $$ y=6 x-3 $$

Differentiate each function $$ y=\left(4 x^{2}+1\right)^{-50} $$

Use the Theorem on Limits of Rational Functions to find each limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 2}(4 x-5) $$

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

Complete each of the following statements. The notation ______ is read 鈥渢he limit as x approaches 5.鈥�

Find \(\frac{d y}{d x}\). $$ y=x^{-8} $$

Differentiate two ways: first, by using the Product Rule; then, by multiplying the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(f(x)=(2 x+5)\left(3 x^{2}-4 x+1\right)\)

Find \(d^{2} y / d x^{2}\) $$ y=\frac{1}{x^{3}} $$

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). $$f(x)=\frac{3}{4} x-2$$

Differentiate two ways: first, by using the Product Rule; then, by multiplying the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(g(x)=(4 x-3)\left(2 x^{2}+3 x+5\right)\)