Rules of Differentiation — Sum, Product and Quotient

Most functions met in practice are built from simple pieces by addition, multiplication and division. Three rules handle these constructions.

Sum / difference rule and constant multiple. The derivative distributes across sums and pulls constants out:

So a polynomial is differentiated term by term.

Product rule. The derivative of a product is not the product of the derivatives. Instead, if $y = u \cdot v$ where $u$ and $v$ are both functions of $x$,

Quotient rule. If $y = \dfrac{u}{v}$ (with $v \neq 0$), then

The order of the two terms in the numerator matters — the denominator's own derivative always comes second, with a minus sign.

Method — choosing a rule:

1. If the function is a sum/difference, differentiate each term separately.
2. If it is a product of two (or more) factors, use the product rule.
3. If it is a genuine ratio that will not simplify, use the quotient rule.
4. Always check first whether algebra removes the need for a rule — e.g. $\dfrac{x^3+x}{x} = x^2 + 1$ is far easier differentiated after cancelling than by the quotient rule.