Partial derivative
High school
Consider a function $f(x, y)$ of two variables. The partial derivative of $f$ with respect to $x$ is the derivative of $f$ found by treating $y$ as a constant.
Undergraduate
Consider a function $f$ of several variables $x_1, \dots, x_n$. The partial derivative $\frac{\partial f}{\partial x_i}$ is the derivative of $f$ with respect to $x_i$ while the other variables are held fixed:
Examples
Worked examples
-
$f(t, x) = t x^2$
: $\frac{\partial f}{\partial t} = x^2$ and $\frac{\partial f}{\partial x} = 2 t x$
Undergraduate
$f$ has two variables, so it has two partial derivatives, and no ordinary derivative.
- $f(x, y) = x^2 + 3y$ : $\frac{\partial f}{\partial x} = 2x$ and $\frac{\partial f}{\partial y} = 3$ Undergraduate
Properties
- Having all partial derivatives at a point does not make a function differentiable there, or even continuous.
- If the second partial derivatives are continuous, then the order of differentiation does not matter: \[\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}\] (Schwarz’s theorem).