CalculusCalculus 1College

Derivatives

Power, product, quotient and chain rules.

The derivative $f'(x)$ is the instantaneous rate of change of $f$ at $x$ , and geometrically it is the slope of the tangent line. It is defined as the limit $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ , but in practice you rarely use the definition, because a handful of rules cover almost every function.

The power rule handles polynomials, the product and quotient rules handle combinations, and the chain rule handles compositions such as $\sin(3x^2)$ . Derivatives describe velocity from position, marginal cost from total cost and growth rates in biology.

Key terms

Power rule: Bring the exponent down and reduce it by one.
$\frac{d}{dx}x^n = nx^{n-1}$
Chain rule: Differentiate the outer function, then multiply by the derivative of the inner.
$\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)$

Key ideas

Power rule: $\frac{d}{dx} x^n = n x^{n-1}$ .
Product rule: $(uv)' = u'v + uv'$ ; quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ .
Chain rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$ .
$\frac{d}{dx} \sin x = \cos x$ , $\frac{d}{dx} e^x = e^x$ , $\frac{d}{dx} \ln x = \frac{1}{x}$ .

Worked example

Differentiate $f(x) = x^2 \sin(3x)$ .

Product rule with $u = x^2$ , $v = \sin(3x)$ .
$u' = 2x$ ; by the chain rule $v' = 3\cos(3x)$ .
$f'(x) = 2x\sin(3x) + 3x^2\cos(3x)$ .

Answer: $2x\sin(3x) + 3x^2\cos(3x)$

Common mistakes

Writing $(uv)' = u'v'$ .
Forgetting the inner derivative in the chain rule.
Differentiating a constant multiple of $x$ to the constant itself plus extra terms.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

Derivatives

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