Partial Derivative Calculator

How to Use the Partial Derivative Calculator

1. Type your function of x, y and z. Use ^ for powers and a space or * between factors.
2. Choose the partial derivative: f_x, f_y, f_z, second order or mixed such as f_xy.
3. Optionally enter a point for x, y and z to evaluate the derivative and gradient.
4. Read the partial derivative, then the value at your point, the gradient and each step.

Enter a function of up to three variables, x, y and z, then choose which partial derivative you want from the list. First-order, pure second-order and mixed partials such as f_xy are all available.

Fill in a point to get the numerical value and the gradient vector at that point. Clear the point boxes if you only want the formula, and use the example buttons to try ready-made functions.

The steps explain which variables are held constant and which differentiation rule applies to each term. The algebra runs in your browser, and each result is compared with a numerical derivative before it is shown.

What a Partial Derivative Means

A partial derivative measures how a multivariable function changes when only one variable moves and all the others stay fixed. For f(x, y), ∂f/∂x is exactly the slope of the surface in the x direction.

In the same way, ∂f/∂y is the slope of the same surface in the y direction. The curly ∂ symbol tells you the function has more than one variable, unlike d in an ordinary single-variable derivative.

The formal definition is a limit, just like an ordinary derivative, except that only one variable gets the small step h. The OpenStax Calculus Volume 3 chapter covers the definition and its geometry in detail.

How to Calculate a Partial Derivative

To differentiate with respect to x alone, treat every other letter as a constant number. That golden rule, hold it constant, turns a partial derivative into an ordinary derivative you already know how to take.

Then apply the usual differentiation rules: the power rule, product rule, quotient rule and chain rule. For f = x²y, the partial with respect to x is 2xy, and with respect to y it is x².

Terms that contain only the other variables simply disappear, because the derivative of a constant is zero. For example, in x² + e^y, the e^y term vanishes when you differentiate with respect to x.

Worked Example: A Function of Two Variables

To find the partial derivative f_x, hold y constant throughout. The first term gives 2xy³, and the chain rule applied to sin(xy) gives y cos(xy), so f_x = 2xy³ + y cos(xy) for every point.

To find f_y, hold x constant and let y vary. The first term gives 3x²y² and the second gives x cos(xy), so f_y = 3x²y² + x cos(xy), using exactly the same chain rule step.

At the point (1, 2), f_x is about 15.16771 and f_y is about 11.58385, so the gradient is about (15.16771, 11.58385). The mixed partial f_xy at the same point is about 21.76526.

• f_xy = 6xy² + cos(xy) − xy sin(xy)
• Differentiating f_y with respect to x gives the same expression, so f_yx = f_xy.

Mixed Partials and Clairaut's Theorem

Second-order partial derivatives come in two kinds, pure and mixed. Pure partials such as f_xx differentiate twice with respect to the same variable, while mixed partials such as f_xy use two different variables.

When the second partial derivatives are continuous, the order of differentiation does not matter, so f_xy = f_yx. This important result is widely known as Clairaut's theorem, or sometimes as Schwarz's theorem on symmetry.

You can test the theorem yourself by choosing f_xy and then f_yx for the same function. In the worked example both give about 21.76526 at the point (1, 2), as the theorem predicts.

The Gradient Vector

The gradient collects all first partial derivatives into one vector, written ∇f. At any point it points in the direction in which the function increases fastest, and its length equals that fastest rate of increase.

The gradient is used in optimization, for example gradient descent in machine learning, which repeatedly steps against the gradient to reduce an error function. It also gives tangent planes and directional derivatives on curved surfaces.

Partial derivatives appear across physics, engineering and economics, from heat flow and wave motion to marginal cost. MIT OpenCourseWare multivariable calculus is a free course that builds on these ideas, including the multivariable chain rule.

Notes and Limits

The calculator supports the variables x, y and z. Other letters, such as a or k, are treated as constants, so a value at a point needs numbers in their place before it is evaluated.

In this tool, ln and log both mean the natural logarithm, base e. Answers can look different from a textbook while being equal, because the same expression can be written in many algebraically equivalent forms.

Implicit functions and piecewise definitions are not supported, so implicit differentiation must be done by hand. If you mistype an expression, an error message appears instead of a result, so check your brackets and operators.

Frequently asked questions