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

Unless otherwise stated, all functions are functions of real numbers (${\textstyle \mathbb {R} }$) that return real values, although, more generally, the formulas below apply wherever they are well defined, including the case of complex numbers (${\textstyle \mathbb {C} }$).

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

Let ${\textstyle c\in \mathbb {R} }$ and ${\textstyle f(x)=c}$. By the definition of the derivative: $${\displaystyle {\begin{aligned}f'(x)&=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}\\&=\lim _{h\to 0}{\frac {(c)-(c)}{h}}\\&=\lim _{h\to 0}{\frac {0}{h}}\\&=\lim _{h\to 0}0\\&=0.\end{aligned}}}$$

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

The derivative of the function at a point is the slope of the line tangent to the curve at the point. The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0.

In other words, the value of the constant function, ${\textstyle y}$, will not change as the value of ${\textstyle x}$ increases or decreases.

For any functions ${\textstyle f}$ and ${\textstyle g}$ and any real numbers ${\textstyle a}$ and ${\textstyle b}$, the derivative of the function ${\textstyle h(x)=af(x)+bg(x)}$ with respect to ${\textstyle x}$ is ${\textstyle h'(x)=af'(x)+bg'(x)}$.