Partial Derivative

For a function of several variables $f(x, y, z, \ldots)$, the partial derivative with respect to $x$ is

$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y, \ldots) - f(x, y, \ldots)}{h},$

treating all other variables as constants. Notation: $\partial$ (rounded "d", read "del") distinguishes from total derivatives.

Example: $f(x, y) = x^2 y + 3y$. Then $\frac{\partial f}{\partial x} = 2xy$ (treating $y$ as constant) and $\frac{\partial f}{\partial y} = x^2 + 3$.

Partial derivatives are the building blocks of multivariable calculus. The gradient $\nabla f = (\partial f/\partial x, \partial f/\partial y, \ldots)$ points in the direction of steepest ascent — the foundation of gradient descent in machine learning. Partial differential equations model heat, waves, fluids, electromagnetism, and quantum mechanics.