User:Melikamp/b-calc

Differentiation Review

1. Write the product function ${\displaystyle f(x)}$ and its rate of change function for ${\displaystyle g(x)=2e^{5}x}$ and ${\displaystyle h(x)=2x^{2}-x}$.

${\displaystyle f(x)=2e^{5}x(2x^{2}-x),\ f'(x)=2e^{5}(2x^{2}-x)+2e^{5}x(4x-1)}$

2. Write the product function ${\displaystyle f(x)}$ and its rate of change function for ${\displaystyle g(x)=0.5\cos(4x)}$ and ${\displaystyle h(x)=2\ln(x)}$.

${\displaystyle f(x)=\cos(4x)\ln(x),\ f'(x)=-4\sin(4x)\ln(x)+{\frac {\cos(4x)}{x}}}$

3. Find ${\displaystyle f'(x)}$ if ${\displaystyle f(x)=(2x+3)e^{-x}}$.

${\displaystyle 2e^{-x}-(2x+3)e^{-x}}$

4. Find ${\displaystyle f'(x)}$ if ${\displaystyle f(x)=2{\sqrt {x^{3}}}\ln(x)}$.

${\displaystyle 3{\sqrt {x}}\ln(x)+2{\sqrt {x}}}$

5. Find ${\displaystyle f'(x)}$ if ${\displaystyle f(x)=(x+2)^{3}(1-x^{2})}$.

${\displaystyle 3(x+2)^{2}(1-x^{2})-2x(x+2)^{3}}$

6. Find ${\displaystyle f'(x)}$ if ${\displaystyle \displaystyle f(x)={\frac {e^{x}}{x}}}$.

${\displaystyle \displaystyle {\frac {xe^{x}-e^{x}}{x^{2}}}}$