Quotient rule
If $f$ and $g$ are differentiable functions, then so is their quotient $f/g$ whenever the denominator is nonzero, and
$\displaystyle \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{\color{blue}{f'(x)}\,g(x) - f(x)\,\color{blue}{g'(x)}}{g(x)^2}$.
Quotient rule in words:
#[Examples][Ejemplos]#
1. \t $\displaystyle \frac{d}{dx}\left[\frac{3x^3 + 7}{4x^2 - x + 4}\right]$ \t #[$\displaystyle {}=\frac{\color{blue}{\text{Deriv of top }} \times \text{ Bottom } - \text{ Top }\times \color{blue}{\text{ Deriv of bottom }}}{\text{Bottom}^2}$][$\displaystyle {}=\frac{\color{blue}{\text{Deriv del Numer }} \times \text{ Denom } - \text{ Numer }\times \color{blue}{\text{ Deriv del denom }}}{\text{Denom}^2}$]#
\\ \t \t $\displaystyle {}= \frac{(\color{blue}{9x^2})(4x^2-x+4) \ \ - \ \ (3x^3 + 7)(\color{blue}{8x-1})}{(4x^2 - x + 4)^2}$
\\ 2. \t $\displaystyle \frac{d}{dx}\left[\frac{|x|}{x^2 + 4}\right]$ \t !2! #[$\displaystyle {}=\frac{\color{blue}{\text{Deriv of top }} \times \text{ Bottom } - \text{ Top }\times \color{blue}{\text{ Deriv of bottom }}}{\text{Bottom}^2}$][$\displaystyle {}=\frac{\color{blue}{\text{Deriv del Numer }} \times \text{ Denom } - \text{ Numer }\times \color{blue}{\text{ Deriv del denom }}}{\text{Denom}^2}$]#
\\ \t \t $\displaystyle {}=\frac{\color{blue}{\frac{|x|}{x}}(x^2 + 4) - |x|\color{blue}{(2x)}}{(x^2+4)^2} \qquad $ \t
\\ \t \t !2! $\displaystyle {}=\frac{|x|\left(x+\frac{4}{x} - 2x\right)}{(x^2+4)^2}$
\\ \t \t $\displaystyle = \frac{|x|\left(-x+\frac{4}{x}\right)}{(x^2+4)^2}$
\\ \t \t $\displaystyle = \frac{|x|\left(4-x^2\right)}{x(x^2+4)^2}$ \t
#[Note][Nota]# In some circumstances, it is helpful to simplify the answer; you should use your best judgment in each situation to decide how much to simplify.