Math basics › Multivariable calculus

Partial derivatives

One variable at a time

∂f/∂x is the derivative with respect to x, treating every other letter as a constant number.

Constants stay put

In ∂/∂x of 5x³y, the 5y is just a coefficient: you get 15x²y. A term with no x in it at all gives 0.

The chain rule still applies

∂/∂y of sin(x + y²) is cos(x + y²) times 2y, the inside's derivative with respect to y.

Second partials

Differentiating twice gives four second partials. The two mixed ones (x then y, y then x) come out equal for the functions you will meet, which makes a handy check.