How Partial Derivatives Work
A partial derivative measures how a multivariable function changes when you vary one variable while freezing the others. The notation โf/โx reads โthe partial of f with respect to xโ โ and every other variable is treated as a plain constant during the differentiation.
The Constant Rule in Action
For f(x, y) = xยฒy + sin(y): differentiating with respect to x treats y as a constant, so โf/โx = 2xy. Differentiating with respect to y treats x as a constant, giving โf/โy = xยฒ + cos(y). The same power, product, and chain rules from single-variable calculus all still apply.
Where They Matter
Partial derivatives power gradient descent in machine learning, heat and wave equations in physics, and marginal analysis in economics. For single-variable differentiation, see our derivative calculator.