Chapter 1

Differentiate each function. \(g(x)=\left(5 x^{2}+4 x-3\right)\left(2 x^{2}-3 x+1\right)\)

Differentiate each function $$ G(x)=\sqrt[3]{2 x-1}+(4-x)^{2} $$

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow-2} \frac{x^{2}-2 x-8}{x^{2}-4} $$

Find \(\frac{d y}{d x}\). $$ y=\frac{7}{x^{3}} $$

Find \(f^{\prime \prime}(x)\) $$ f(x)=\left(2 x^{2}-3 x+1\right)^{10} $$

Find an equation of the tangent line to the graph of \(f(x)=4-x^{2}\) at (a) (-1,3) (b) (0,4) (c) (5,-21) .

Differentiate each function. \(f(x)=\left(3 x^{2}-2 x+5\right)\left(4 x^{2}+3 x-1\right)\)

Differentiate each function $$ g(x)=\sqrt{x}+(x-3)^{3} $$

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 1} \frac{x^{2}+5 x-6}{x^{2}-1} $$

Find \(\frac{d y}{d x}\) $$ y=\frac{3 x}{4} $$

Find \(f^{\prime \prime}(x)\) $$ f(x)=\sqrt[4]{\left(x^{2}+1\right)^{3}} $$

Find \(f^{\prime}(x)\) for \(f(x)=m x+b\).

Differentiate each function. \(y=\frac{5 x^{2}-1}{2 x^{3}+3}\)

Differentiate each function $$ f(x)=-5 x(2 x-3)^{4} $$

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 2} \frac{3 x^{2}+x-14}{x^{2}-4} $$

Find \(\frac{d y}{d x}\). $$ y=\frac{4 x}{5} $$

Find \(f^{\prime \prime}(x)\) $$ f(x)=\sqrt[3]{\left(x^{2}-1\right)^{2}} $$

Find \(f^{\prime}(x)\) for \(f(x)=a x^{2}+b x\).

Differentiate each function. \(y=\frac{3 x^{4}+2 x}{x^{3}-1}\)

Differentiate each function $$ f(x)=-3 x(5 x+4)^{6} $$

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow-3} \frac{2 x^{2}-x-21}{9-x^{2}} $$

Find each derivative. $$ \frac{d}{d x}\left(\sqrt[4]{x}-\frac{3}{x}\right) $$

Find \(y^{\prime \prime}\) $$ y=x^{3 / 2}-5 x $$

Differentiate each function. \(F(x)=\left(-3 x^{2}+4 x\right)(7 \sqrt{x}+1)\)

Differentiate each function $$ F(x)=(5 x+2)^{4}(2 x-3)^{8} $$

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 2} \frac{x^{3}-8}{2-x} $$

Find each derivative. $$ \frac{d}{d x}\left(\sqrt[3]{x}+\frac{4}{\sqrt{x}}\right) $$

Find \(y^{\prime \prime}\) $$ y=x^{2 / 3}+4 x $$

Differentiate each function $$ g(x)=(3 x-1)^{7}(2 x+1)^{5} $$

Differentiate each function. \(G(x)=(8 x+\sqrt{x})\left(5 x^{2}+3\right)\)

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1} $$

Find each derivative. $$ \frac{d}{d x}\left(-2 \sqrt[3]{x^{5}}\right) $$

Find \(y^{\prime \prime}\) $$ y=\left(x^{3}-x\right)^{3 / 4} $$

Differentiate each function $$ f(x)=x^{2} \sqrt{4 x-1} $$

Differentiate each function. \(g(t)=\frac{t}{3-t}+5 t^{3}\)

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{x-25} $$

Find each derivative. $$ \frac{d}{d x}\left(-\sqrt[4]{x^{3}}\right) $$

Find \(y^{\prime \prime}\) $$ y=\left(x^{4}+x\right)^{2 / 3} $$

Differentiate each function $$ f(x)=x^{3} \sqrt{5 x+2} $$

Differentiate each function. \(f(t)=\frac{t}{5+2 t}-2 t^{4}\)

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 9} \frac{9-x}{\sqrt{x}-3} $$

Based on data from Major League Baseball, the average price of a ticket to a major league game can be approximated by $$ p(x)=0.06 x^{3}-0.5 x^{2}+1.64 x+24.76 $$ where \(x\) is the number of years after 2008 and \(p(x)\) is in dollars. (Source: Based on data from www.teammarketing .com.) a) Find \(p(4)\). b) Find \(p(6)\). c) Find \(p(6)-p(4)\) d) Find \(\frac{p(6)-p(4)}{6-4}\). What rate of change does this represent?

Find each derivative. $$ \frac{d}{d x}\left(5 x^{2}-7 x+3\right) $$

Draw a graph that is continuous, but not differentiable, at \(x=3\).

Find \(y^{\prime \prime}\) $$ y=3 x^{4 / 3}-x^{1 / 2} $$

Differentiate each function $$ F(x)=\sqrt[4]{x^{2}-5 x+2} $$

Differentiate each function. \(G(x)=(5 x-4)^{2}\)

The initial substitution of \(x=a\) yields the form \(0 / 0 .\) Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. $$ \lim _{x \rightarrow 2} \frac{x^{2}+3 x-10}{x^{2}-4 x+4} $$

The amount of money, \(A(t),\) in a savings account that pays \(6 \%\) interest, compounded quarterly for \(t\) years, when an initial investment of \(\$ 2000\) is made, is given by $$ A(t)=2000(1.015)^{4 t} $$ a) Find \(A(3)\). b) Find \(A(5)\) c) Find \(A(5)-A(3)\). d) Find \(\frac{A(5)-A(3)}{5-3}\). What rate of change does this represent?

Find each derivative. $$ \frac{d}{d x}\left(5 x^{2}-7 x+3\right) $$