Derivative Quotient Rule

The quotient rule gives the derivative of a quotient of functions.

Statement

If $f$ and $g$ are functions of $x$, then

$$\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{g\cdot \frac{d}{dx}f-f\cdot \frac{d}{dx}g}{g^2}$$

Derivation

The derivation of the quotient rule using the Product Rule and Chain Rule.

$$\begin{array}{rl}\frac{d}{dx}\frac{f(x)}{g(x)}& =\frac{d}{dx}\left[f(x)\cdot \frac{1}{g(x)}\right]\\ & =f^{\prime }(x)\cdot \frac{1}{g(x)}+f(x)\cdot \frac{d}{dx}\left[g(x{)}^{-1}\right]\\ & =f^{\prime }(x)\cdot \frac{1}{g(x)}+f(x)\cdot \left[-1\cdot g(x{)}^{-2}\cdot g^{\prime }(x)\right]\\ & =\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{g(x{)}^2}\end{array}$$