Differentiation formulas and rules

Derivatives of elementary functions

These formulas follow from the definition in the previous lesson. They’re worth knowing by heart.

Function $f(x)$	Derivative $f'(x)$	Notes
$c$ (constant)	$0$
$x^n$	$n x^{n-1}$
$\sqrt{x}$	$\dfrac{1}{2\sqrt{x}}$	for $x > 0$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$e^x$	$e^x$
$\ln x$	$\dfrac{1}{x}$	for $x > 0$

Differentiation rules

For differentiable functions $f$ and $g$ and a constant $c$:

A common mistake: the derivative of a product is not the product of the derivatives.

Example 1: a polynomial

Differentiate each term separately: $(4x^3)' = 12x^2$, $(-5x)' = -5$, and the derivative of the constant $2$ is $0$.

Example 2: a product

Example 3: a quotient

for $x \neq -1$.