Partial derivative calculator

What a partial derivative measures

A function of one variable has one rate of change. A function of two has a different rate in every direction, so "how fast does \(f\) change?" needs a direction attached to it. The partial derivatives answer the simplest version of that question: how fast \(f\) changes if you move parallel to an axis.

For \(f(x, y)\), the partial \(\partial f/\partial x\) is taken by treating \(y\) as a constant and differentiating normally. Everything you already know still applies — the product rule, the chain rule, the power rule. The only new instruction is which symbols to freeze.

Reading the notation

• \(\partial f/\partial x\), \(f_x\) and \(\partial_x f\) all mean the same thing.
• The curly \(\partial\) is used instead of \(d\) precisely to signal that other variables are held fixed.
• \(f_{xy}\) means differentiate with respect to \(x\) first, then \(y\) — reading order runs left to right.

Worked example in two variables

Take \(f(x, y) = x^{2}y^{3} + \sin(x)\,y\).

With respect to x

Hold \(y\) constant. The first term carries the constant factor \(y^{3}\), so the power rule gives \(2xy^{3}\). In the second term \(y\) is a constant multiplier, so the trigonometric rule gives \(y\cos x\). Together:

\[\frac{\partial f}{\partial x} = 2xy^{3} + y\cos x\]

With respect to y

Now \(x\) is frozen. The first term has the constant factor \(x^{2}\), giving \(3x^{2}y^{2}\). The second term is a constant \(\sin(x)\) times \(y\), which differentiates to \(\sin x\). Together:

\[\frac{\partial f}{\partial y} = 3x^{2}y^{2} + \sin x\]

Change the variable control above and watch the steps swap over. Same engine, same rules — only the frozen symbols differ.

Second-order and mixed partials

Differentiating twice gives four second-order partials: \(f_{xx}\), \(f_{yy}\), and the two mixed ones \(f_{xy}\) and \(f_{yx}\). Set the order control to 2 for \(f_{xx}\) or \(f_{yy}\) directly.

For the example above, \(f_{xx} = 2y^{3} - y\sin x\) and \(f_{yy} = 6x^{2}y\). Both mixed partials come to \(6xy^{2} + \cos x\) — Clairaut's theorem in action: when the second partials are continuous, the order of differentiation does not matter.

The gradient

Collecting the partials into a vector gives the gradient, \(\nabla f = \langle f_x, f_y \rangle\). It points in the direction of steepest increase, and its length is how steep that is. Every directional derivative is a dot product with it, which is why the directional derivative calculator begins by computing exactly these partials.

Total derivative versus partial derivative

If \(x\) and \(y\) are themselves functions of \(t\), then \(f\) changes for two reasons at once and the partials alone do not capture it. The multivariable chain rule adds the contributions:

\[\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}\]

This is also the machinery behind error propagation in physics: the uncertainty in a computed quantity combines the partial derivatives with respect to each measured input.

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