Partial Derivative Calculator

Step-by-step working

1Sum rule
$\frac{\partial}{\partial x}\left[x^{2}y+\sin\left(xy\right)\right]$
$\frac{\partial}{\partial x}\left[x^{2}y\right]+\frac{\partial}{\partial x}\left[\sin\left(xy\right)\right]$
2Constant multiple rule
$\frac{\partial}{\partial x}\left[x^{2}y\right]$
$y\cdot \frac{\partial}{\partial x}\left[x^{2}\right]$
3Power rule
$\frac{\partial}{\partial x}\left[x^{2}\right]$
$2x$
4Chain rule
$\frac{\partial}{\partial x}\left[\sin\left(xy\right)\right]$
$\cos\left(xy\right)\cdot \frac{\partial}{\partial x}\left[xy\right]$
Outer function \sin u, inner u = xy.
5Constant multiple rule
$\frac{\partial}{\partial x}\left[xy\right]$
$y\cdot \frac{\partial}{\partial x}\left[x\right]$
6Simplify
$y2x^{1}+\cos\left(xy\right)y$
$2xy+y\cos\left(xy\right)$

Worked solution: partial derivative of x^2y + sin(xy) with respect to x

Worked steps

1Sum rule
$\frac{\partial}{\partial x}\left[x^{2}y+\sin\left(xy\right)\right]$
$\frac{\partial}{\partial x}\left[x^{2}y\right]+\frac{\partial}{\partial x}\left[\sin\left(xy\right)\right]$
2Constant multiple rule
$\frac{\partial}{\partial x}\left[x^{2}y\right]$
$y\cdot \frac{\partial}{\partial x}\left[x^{2}\right]$
3Power rule
$\frac{\partial}{\partial x}\left[x^{2}\right]$
$2x$
4Chain rule
$\frac{\partial}{\partial x}\left[\sin\left(xy\right)\right]$
$\cos\left(xy\right)\cdot \frac{\partial}{\partial x}\left[xy\right]$
Outer function \sin u, inner u = xy.
5Constant multiple rule
$\frac{\partial}{\partial x}\left[xy\right]$
$y\cdot \frac{\partial}{\partial x}\left[x\right]$
6Simplify
$y2x^{1}+\cos\left(xy\right)y$
$2xy+y\cos\left(xy\right)$

Answer

\frac{\partial}{\partial x}\left[x^{2}y+\sin\left(xy\right)\right]=2xy+y\cos\left(xy\right)

About Partial Derivative Calculator

A partial derivative measures how a function of several variables changes when only one of them moves. This page opens with f(x, y) = x^2 y + sin(xy) differentiated with respect to x, which gives 2xy + y cos(xy). It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

To take a partial derivative you treat every other letter as a number. In x^2 y the factor y is a constant multiplier, so the derivative with respect to x is 2xy. In sin(xy) the chain rule applies, and the inner derivative of xy with respect to x is y.

Change "With respect to" to y to get the other partial derivative, x^2 + x cos(xy). The graph is drawn only for functions of a single variable, so it is hidden here.

How to use Partial Derivative Calculator

Type the function in the Function box, for example x^2*sin(x), or tap keys on the key pad. The preview underneath shows how it was read.
Choose the order: 1st for the ordinary derivative, up to 5th for higher derivatives.
To differentiate with respect to another letter, change "With respect to". Other letters are then treated as constants, which gives a partial derivative.
For an equation such as x^2 + y^2 = 25, switch to Implicit and type both sides with an equals sign.
Enter a point such as 2 or pi/4 to see the value, the slope and the tangent line there, drawn on the graph.
Read the steps under the result to see which rule was used at each stage, and copy the answer with the copy button.

Features

First to fifth derivatives of a function of one variable.
Partial derivatives: choose any variable and the others are held constant.
Implicit differentiation of an equation in x and y, solved for dy/dx.
Value, slope and tangent line equation at a point you choose, with the tangent drawn on the graph.
Step-by-step working that names the power, product, quotient and chain rules and the standard derivative used.
A live preview and a math key pad, so input is easy on a phone.
A zoomable, draggable graph of the function and its derivative that you can save as a PNG.
Clear errors: input the calculator cannot read is flagged with the position of the problem.

Worked examples

f(x, y) = x^2*y + sin(x*y), with respect to x gives 2*x*y + y*cos(x*y)
y is held constant.

Privacy: how it works

The derivative is worked out by code running in your browser tab. Your functions are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

Use * between factors when you can: 2*x is always read as a product.
To check a result, integrate it with the integral calculator and see if you get the original function back, up to a constant.
A tangent line is the best straight-line approximation to the function near the point, which is why it is shown next to the graph.
For the nth derivative of a function that repeats, such as sin(x), pick the order and watch the pattern: sin, cos, -sin, -cos.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Your functions are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Your functions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

What is the partial derivative of x^2 y + sin(xy) with respect to x?

It is 2xy + y cos(xy). The first term gives 2xy by the power rule with y as a constant, and the second uses the chain rule.

How is a partial derivative different from an ordinary one?

An ordinary derivative is for a function of one variable. A partial derivative picks one variable of a function of several and treats the rest as constants, written with a curly d.

How do I get the partial derivative with respect to y?

Type y in "With respect to". For this function the answer becomes x^2 + x cos(xy).