partial derivative

Imagine you are standing on a hill whose height depends on two things: how far east you are and how far north you are. Ask a narrower question: if I face due east and take a step, while not moving north at all, how steeply does the ground rise? That single-direction slope is a partial derivative. You freeze every variable except one and measure the rate of change in just that one direction.

For a function f(x, y) of several inputs, the partial derivative with respect to x is written ∂f/∂x. You compute it exactly like an ordinary derivative, but you treat every other variable as a fixed constant. For example, if f(x, y) = x^2 y + y^3, then ∂f/∂x = 2xy (treating y as a number), and ∂f/∂y = x^2 + 3y^2 (treating x as a number). The curly symbol ∂ is read 'partial' and signals that other variables are being held still.

A common confusion: a partial derivative tells you the slope along one chosen axis only, not the steepest slope overall. On a real surface the ground may climb much faster in some diagonal direction than along either axis. To capture slopes in every direction you combine the partials into the gradient, and a single number like ∂f/∂x never tells the whole story by itself.