When a function depends on several variables — like $f(x, y) = x^2 + xy$ — you can ask: how fast does it change if I move in just the $x$-direction? Or just the $y$-direction?

A partial derivative answers: what’s the rate of change in one direction, while holding everything else fixed?

The idea

Imagine standing on a hillside. The elevation $h(x, y)$ depends on your east-west position ($x$) and your north-south position ($y$).

• $\frac{\partial h}{\partial x}$: How steep is the hill if you walk east? (Hold $y$ fixed, vary $x$.)
• $\frac{\partial h}{\partial y}$: How steep is the hill if you walk north? (Hold $x$ fixed, vary $y$.)

The symbol $\partial$ (a curly d) distinguishes partial derivatives from ordinary derivatives. It means “change this variable, freeze the others.”

How to compute them

Take the partial derivative the same way you’d take an ordinary derivative — just treat every other variable as a constant.

For $f(x, y) = x^2 + xy$:

$$\frac{\partial f}{\partial x} = 2x + y \qquad \text{(treat } y \text{ as a constant)}$$

$$\frac{\partial f}{\partial y} = x \qquad \text{(treat } x \text{ as a constant)}$$

Why they matter here

Divergence and curl are both built from partial derivatives:

$$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} \qquad \nabla \times \mathbf{F} = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}$$

If you can take a derivative of a single-variable function, you can take partial derivatives — you just do it one variable at a time.