The derivative of a function is a second function showing the rate of change of the dependent variable compared to the independent variable. It can be thought of as a graph of the slope of the function from which it is derived. The process of finding a derivative is called differentiation. The derivative of y with respect to x (denoted as y') is equal to

${\displaystyle y' = \lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x} = \frac{\mathrm{d}x}{\mathrm{d}y}\,}$

The inverse of differentiation is integration.

Multivariable extensions

For multivariable scalar fields, either a gradient (a vector field representing the direction of greatest change) or directional derivative (a scalar representing the rate of change in a particular direction) can be taken. The two equivalents to derivation over a scalar field are divergence and curl.

Derivatives in physics

${\displaystyle x'(t) = v(t)}$

${\displaystyle v'(t) = a(t)}$

${\displaystyle W'(d) = F(d)}$

${\displaystyle W'(t) = P(t)}$

${\displaystyle \nabla \cdot V = \vec{E}}$

f(x)=f'(X)

See also

• Table of derivatives
• Integration
• Derivative on Math wiki