The Product Rule of Derivatives
The Formula
When is the Product Rule used? 
How does the Product Rule Work ? 

Why Does this work? 
Notes
If f(x) and g(x) are differentiable at x then h(x)=f(x)g(x) is also differentiable at x.
 $f(x)g'(x)+g(x)f'(x)$
Example 

Practice Problems
Apply the product rule to each expression below. Use our derivative of functions table to help.
Problem 1) $tan(x)ln(x)$ (hint) 
$tan(x)\frac{1}{x}+ln(x)sec^2x$ or $\frac{tan(x)}{x}+ln(x)sec^2x$ 
Problem 2) $e^xsin^{1}x$ (hint) 
$e^x\frac{1}{\sqrt{1x^2}}+e^xsin^{1}x$ or $e^x(\frac{1}{\sqrt{1x^2}}+sin^{1}x)$ 
Problem 3)
$2x^5cos(x)$ (hint) 
$2x^5(sin(x))+10x^4cos(x)$ or $2x^4(5cos(x)xsin(x))$ 

What if we have more than 2 functions? 
If f(x), g(x) and s(x) are differentiable functions with known derivatives then: $$h(x)= \color{red}{f(x)} \color{blue}{g(x)}\color{green}{s(x)} $$ $$h'(x)=\color{red}{f(x)}\color{blue}{g(x)}\color{green}{s \boldsymbol '(x)} + \color{red}{f(x)}\color{blue}{g\boldsymbol '(x)}\color{green}{s(x)} + \color{red}{f\boldsymbol'(x)}\color{blue}{g(x)}\color{green}{s(x)}$$Examples
$xln(x)sin(x)$ (hint)
Show Derivative 
$xln(x)cos(x)+x(\frac{1}{x})sin(x)+ln(x)sin(x)$
or
$ xln(x)cos(x)+sin(x)(ln(x)+1) $
or
$ xln(x)cos(x)+sin(x)(ln(x)+1) $
$x^3e^xtan^{1}x$
(hint)
Show Derivative 
$x^3e^x(\frac{1}{1+x^2})+x^3e^xtan^{1}x+3x^2e^xtan^{1}x$
or
$x^2e^x(\frac{x}{1+x^2} +xtan^{1}x+3tan^{1}x)$
or
$x^2e^x(\frac{x}{1+x^2} +xtan^{1}x+3tan^{1}x)$
The Quotient Rule