Partial Derivative Calculator
Free partial derivative calculator. Differentiate f(x, y, z) in one variable while the others stay constant, with steps.
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<iframe src="https://derivativecalculatorai.com/partial-derivative-calculator" title="Derivative Calculator AI" width="100%" height="640" style="border:0"></iframe>A partial derivative asks how a function changes when only one input moves. In f(x, y) = x^2 y + sin(y), ∂/∂x treats y as a constant, so sin(y) drops out and the x^2 y term becomes 2 x y. ∂/∂y treats x as a constant, so x^2 stays in front of y and the sine contributes cos(y).
That is the whole rule. Ordinary power, product, and chain rules still apply to the active variable. A term with no active variable is a constant for that derivative and contributes 0. Mixed seconds, such as ∂²f/∂x∂y, are just one partial followed by another.
Enter an expression in x, y, and z. Pick the variable. The result is the symbolic partial, not a numeric slope at a point. For a function of x alone, the partial with respect to x matches the ordinary derivative.
Rule and examples
Frequently asked questions
What is a partial derivative?
It is the derivative with respect to one variable while every other variable is held constant.
How is a partial different from an ordinary derivative?
An ordinary derivative has one independent variable. A partial is the same operation after the other inputs are frozen.
What does ∂f/∂y mean for f(x, y)?
Differentiate in y and treat x as a constant coefficient or as part of a constant term.
Can I differentiate with respect to z?
Yes. Choose z. If the expression does not contain z, the partial is 0.