Partial Derivative Calculator

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A partial derivative measures how a function of several variables changes when one variable changes and the others stay fixed. For \(f = x^{2}y + \sin(xy)\), \(f_{x} = 2xy + y\cos(xy)\): treat y as a constant and differentiate with respect to x.

FreeStep by stepSecond partials

Function f

\(f = x^{2} y + \sin\left(x y\right)\)Use x, y and z. Other letters are treated as constants. Write powers with ^ and products with * or a space.

Derivative

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Give a value for every variable.

Try:

Partial derivative

\(\displaystyle \frac{\partial f}{\partial x} = 2x y + y \cos\left(x y\right)\)

Rate of change of f as x changes, holding y fixed.

Function	\(f = x^{2} y + \sin\left(x y\right)\)
Notation	\(f_{x} = \frac{\partial f}{\partial x}\)
Value at (1, 2)	\(2\cos\left(2\right) + 4 \approx 3.16771\)

Show steps

Treat y as constant and differentiate with respect to x, one term at a time:
\(\frac{\partial}{\partial x}\left[x^{2} y\right] = 2x y\) (power rule with a constant multiple)
\(\frac{\partial}{\partial x}\left[\sin\left(x y\right)\right] = y \cos\left(x y\right)\) (chain rule (sin))

How to use the partial derivative calculator

Enter f using x, y and z (for example x^2 y + sin(x y)).
Choose the derivative: fx, fy, a second partial such as fxx or fxy, or the whole gradient.
Optionally give a point like x=1, y=2 to get the value there.
Press Differentiate. Each term is differentiated on its own line, with the rule used.

How to find a partial derivative

To find \(\frac{\partial f}{\partial x}\), treat every variable except x as a constant and use the ordinary rules of differentiation.

Example: \(f(x,y) = x^{3}y^{2} – 4xy + 7\).

\(f_{x} = 3x^{2}y^{2} – 4y\) (y is a constant)
\(f_{y} = 2x^{3}y – 4x\) (x is a constant)
\(f_{xy} = \frac{\partial}{\partial y}\left(3x^{2}y^{2} – 4y\right) = 6x^{2}y – 4\)

Notation

Meaning	Notation
Partial with respect to x	\(f_{x} = \frac{\partial f}{\partial x}\)
Second partial in x	\(f_{xx} = \frac{\partial^{2} f}{\partial x^{2}}\)
Mixed partial: x first, then y	\(f_{xy} = \frac{\partial^{2} f}{\partial y\,\partial x}\)
Gradient	\(\nabla f = \left\langle f_{x}, f_{y}, f_{z} \right\rangle\)

Clairaut’s theorem: when the mixed partials are continuous, \(f_{xy} = f_{yx}\). The calculator shows both orders so you can see it.

The gradient

The gradient \(\nabla f\) points in the direction in which f increases fastest, and its length is that fastest rate. For \(f = x^{2} + y^{2} + z^{2}\) at (1, 2, 2), \(\nabla f = \langle 2, 4, 4 \rangle\) with length 6.

Common mistakes

Forgetting the chain rule: \(\frac{\partial}{\partial x}\sin(xy) = y\cos(xy)\), not \(\cos(xy)\).
Differentiating the “constant” variable: in \(f_{x}\), \(y^{2}\) has derivative 0.
Mixing up the order notation for \(f_{xy}\) and \(\frac{\partial^{2} f}{\partial x\,\partial y}\).

Partial derivatives practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What is a partial derivative?

It is the derivative of a multivariable function with respect to one variable while all the other variables are held constant.

How do you calculate f<sub>xy</sub>?

Differentiate f with respect to x first, then differentiate that result with respect to y.

What does the gradient tell you?

The gradient vector points in the direction of steepest increase of the function, and its length is the rate of increase in that direction.

Can I use letters other than x, y and z?

Yes. Other letters such as a or k are treated as constants, so you can find partial derivatives of formulas with parameters.

How is the answer checked?

The calculator compares the result with a numerical derivative (a central difference) at several points before showing it.

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