# What is the partial derivative of f?

Finds partial derivatives, checked numerically.

- Page: https://www.acalculator.org/math/partial-derivative-calculator
- JSON spec: https://www.acalculator.org/math/partial-derivative-calculator.json
- Version: 4ce37a9e1b1d

## Default answer

Example with the default inputs (Function f(x, y, …) x^2 - 3x y + 2y^2 - 4x + 5y - 12, With respect to x): The partial derivative of x^2 - 3x y + 2y^2 - 4x + 5y - 12 with respect to x is 2x - 3y - 4.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(x, y, …) | The function, in one or more letters. |
| wrt | With respect to | Letters in order: x for ∂f/∂x, xy for f_xy. |
| at | Value at (optional) | A point: x = 1, y = 2 (or 1, 2). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| derivative | Partial derivative | The partial derivative. |
| value | Value at the point | Its value at the point. |

## Method

Each step is a CAS derivative checked at 15 points.

## Assumptions

- Radians.

## Worked examples

1. f = x^2 - 3x y + 2y^2 - 4x + 5y - 12, wrt = y, at = 1, 2 gives derivative = -3x + 4y + 5, value = 10. Source: OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 4.3, Ex. 4.15.
2. f = x e^(-3y) + sin(2x - 5y), wrt = xy gives derivative = 10 sin(2x - 5y) - 3/e^(3y). Source: OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 4.3, Ex. 4.19.

## FAQ

### What is a partial derivative?

A partial derivative of a function of several variables is its rate of change in one variable while the others stay fixed. For f(x, y) = x²y, the partial derivative with respect to x treats y as a constant: ∂f/∂x = 2xy. With respect to y it treats x as a constant: ∂f/∂y = x².

### How do I get a second or mixed partial derivative?

Type the letters in the order you differentiate. xx gives ∂²f/∂x², the derivative with respect to x twice. xy gives f_xy: first with respect to x, then with respect to y, which is ∂²f/∂y∂x. For most functions (when the second partials are continuous) f_xy and f_yx are equal, by Clairaut’s theorem.

### Which letters can I use?

Any lowercase letters except e, which is Euler’s number: x, y, z, t, u and so on. Letters you do not differentiate by are treated as constants. A letter that does not appear in f gives a partial derivative of 0.

### How is the answer checked?

At 15 fixed points in all the letters, the page measures the slope of f numerically, with a difference quotient taken just before and after each point, and compares it with the formula. They must agree to about 1 part in a million. If they do not, the page says "No verified answer" instead of showing a formula.

### How do I evaluate the partial derivative at a point?

Type the point in "Value at", either as x = 1, y = 2 or as 1, 2 (the numbers go with the letters in alphabetical order). For f(x, y) = x² − 3xy + 2y² − 4x + 5y − 12, ∂f/∂y = −3x + 4y + 5, which is 10 at (1, 2).

### Why does my answer look different from the book’s?

The same partial derivative can be written in several ways: −3e^(−3y) and −3/e^(3y) are equal, and so are (2xy − 2)cos(u) and 2(xy − 1)cos(u). The page writes powers of the letters first, highest degree first, but it does not factor or expand the answer.

## Sources

- OpenStax, Calculus Volume 3, section 4.3 Partial Derivatives: https://openstax.org/books/calculus-volume-3/pages/4-3-partial-derivatives