Differentiation rules

This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.

Elementary rules of differentiation

Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined — including the case of complex numbers (C).

Constant term rule

For any value of $c$, where $c\in \mathbb {R}$, if $f(x)$ is the constant function given by $f(x)=c$, then ${\frac {df}{dx}}=0$.

Proof

Let $c\in \mathbb {R}$ and $f(x)=c$. By the definition of the derivative,

${\begin{aligned}f'(x)&=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}\\&=\lim _{h\to 0}{\frac {(c)-(c)}{h}}\\&=\lim _{h\to 0}{\frac {0}{h}}\\&=\lim _{h\to 0}0\\&=0\end{aligned}}$

This shows that the derivative of any constant function is 0.

Differentiation is linear

Main article: Linearity of differentiation

For any functions $f$ and $g$ and any real numbers $a$ and $b$, the derivative of the function $h(x)=af(x)+bg(x)$ with respect to $x$