Rules of differentiation

Computing every derivative from the limit definition would be exhausting. A handful of rules let you differentiate almost any formula by inspection.

The power rule

For any exponent $n$ ,

\frac{d}{d x} x^{n} = n x^{n - 1} .

So the derivative of $x^{3}$ is $3 x^{2}$ , and of $x^{1 / 2}$ is $\frac{1}{2} x^{- 1 / 2}$ . This one rule covers a huge range of functions.

A constant does not change, so its derivative is zero. A constant multiplier rides along: $\frac{d}{d x} c f (x) = c f' (x)$ . And derivatives split across addition: $\frac{d}{d x} big (f + g big) = f' + g'$ . Together these let you differentiate any polynomial term by term. The derivative of $5 x^{3} - 2 x + 7$ is $15 x^{2} - 2$ .

The product rule

The derivative of a product is not the product of the derivatives. Instead,

(f g)' = f' g + f g' .

For $x^{2} \sin x$ , that gives $2 x \sin x + x^{2} \cos x$ .

The quotient rule

For a ratio,

(\frac{f}{g})' = \frac{f' g - f g'}{g^{2}} .

The chain rule

This is the most important rule of all: it differentiates a composition, a function inside a function. If $y = f (g (x))$ , then

\frac{d y}{d x} = f' (g (x)) \cdot g' (x) .

Differentiate the outer function, keep the inner one intact, then multiply by the derivative of the inner. For $(3 x^{2} + 1)^{5}$ , the outer power gives $5 (3 x^{2} + 1)^{4}$ and the inner gives $6 x$ , so the derivative is $30 x (3 x^{2} + 1)^{4}$ . Master the chain rule and most derivatives fall out in a line or two.

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Rules of differentiation

The power rule

Constants and sums

The product rule

The quotient rule

The chain rule