STEM Calculators

Partial Derivative Calculator

Math Calculus • Multivariable Calculus

Written by STEM Calculators Team Published Updated
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1. Partial Derivative Calculator
Computes partial derivatives \(\partial f/\partial x\), \(\partial f/\partial y\) and higher-order mixed derivatives (up to order 4), treating the other variable(s) as constant. Includes a zoomable, pannable contour plot and an optional Clairaut (mixed-equality) check.
Inputs
Variables: x, y. Constants: pi, e. Functions: sin cos tan asin acos atan sqrt abs exp ln/log. Use ^ for powers. Implicit multiplication like 2x is allowed.
Mixed mode lets you build \(\partial^k f / (\partial x\,\partial y\,\dots)\). Repeated mode is for \(\partial^n/\partial x^n\) or \(\partial^n/\partial y^n\).
Enter the denominator order, e.g. x y means \(\dfrac{\partial^2 f}{\partial x\,\partial y}\). Computed as: differentiate w.r.t \(y\) first, then \(x\) (rightmost first), matching operator composition.
order: 2
Uses symbolic derivative if available; otherwise uses central-difference numeric partials.
Click a preset to load & compute.
Ready
Contour plot
Drag to pan • wheel/pinch to zoom. Plot shows a heatmap + contours of \(f(x,y)\). Optional point marker + gradient arrows.
Higher = smoother, slower.
x: 0, y: 0, zoom(px/unit): 60
Results & steps
Click Fill example or press Calculate.

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Frequently Asked Questions

What does a partial derivative mean in multivariable calculus?

A partial derivative measures how f(x,y) changes when one variable changes while the other variable is treated as constant. For example, ∂f/∂x differentiates with respect to x while holding y fixed.

How do I enter a mixed partial like ∂^2 f/(∂x ∂y) in the calculator?

Use Mixed derivative mode and build the denominator order with x and y. The calculator applies the rightmost derivative first, so ∂^2 f/(∂x ∂y) means differentiate with respect to y first, then x.

Why might ∂^2 f/(∂x ∂y) not equal ∂^2 f/(∂y ∂x)?

Clairaut's theorem states mixed partials are equal when the second partial derivatives are continuous near the point of interest. If smoothness conditions fail, the two mixed partials can differ, and the calculator can check equality for second-order mixed cases.

Does the tool use symbolic differentiation or numerical approximation?

It uses symbolic differentiation when available to produce an exact expression. If a symbolic form is not available, it falls back to central-difference numerical partial derivatives for numeric results.

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