The Quotient Rule

The derivative of the quotient is not the quotient of the derivatives.

We write, briefly, $\displaystyle\left(\frac{f}{g}\right)'=\frac{gf'-fg'}{g^2}=\displaystyle\frac{\text{lo de hi}-\text {hi de lo}}{\text{lolo}}$, where hi=numerator, lo=denominator, and de=differentiate. 
The important thing to remember here is that unlike the product rule, where $f'g+fg'=fg'+f'g$ and the order doesn't matter, $gf'-fg'\not = fg'-gf'$.  Lo comes first!

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