Product and quotient rules
Rates of change of products and quotients split in ways intuition gets wrong, so the rules are worth memorizing exactly.
Product rule
The derivative of a product f times g is f' g plus f g'. Each factor's own change must be accounted for, which is why the derivative is not simply the product of the derivatives. Both orders appear because multiplication is symmetric.
(fg)' = f'g + fg'Quotient rule
For a quotient f over g, the derivative is f' g minus f g', all divided by g squared. The minus sign tracks the denominator's growth suppressing the ratio. A memory aid: low d-high minus high d-low, over low squared.
\left( \frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}Worked example
Differentiate f(x) = x^2 e^x using the product rule with f = x^2 and g = e^x. The result is 2x e^x + x^2 e^x, which factors to x e^x (x + 2). For a quotient, the derivative of (x^2 + 1)/x is (2x \cdot x - (x^2+1) \cdot 1)/x^2, which simplifies to (x^2 - 1)/x^2.
(x^2 e^x)' = 2x e^x + x^2 e^xFlashcard
Product rule for derivatives
Check your understanding
What is the derivative of h(x) = x^2 \ln x for x > 0?