The Quotient Rule

The derivative of a quotient is not the derivative of the numerator divided by the derivative of the denominator. The video below shows this with an example. Instead,

The quotient rule can be derived from the product rule. If we write $\displaystyle f(x) = g(x)\cdot \frac{f(x)}{g(x)}$, then the product rule says that $$ f'(x) = \left ( g(x) \cdot\frac{f(x)}{g(x)} \right )', $$ i.e., $$ f'(x)= g'(x) \frac{f(x)}{g(x)} + g(x) \left ( \frac{f(x)}{g(x)} \right )'. $$ Solving for $\left( \frac{f(x)}{g(x)} \right )'$ gives $$ \left ( \frac{f(x)}{g(x)} \right )' = \frac{f'(x) - g'(x)\frac{f(x)}{g(x)}}{g(x)} = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}.$$

The most common mistake with this rule is mix up the signs of the $g f'$ and $f g'$ terms. If $f(x)$ and $g(x)$ are positive, then increasing the numerator $f(x)$ will increase the ratio $f/g$, so the $g f'$ term must be positive. However, increasing the denominator $g(x)$ will decrease the ratio, so the $f g'$ term must be negative.