Partial Derivative Calculator

Differentiate a multivariable function with respect to one variable while holding the others fixed.

Drag & drop or upload an image or PDF

Enter a partial derivative, such as partial d/dx of x^2 y + sin(xy)

Try an example

Change one input and hold the others still

A partial derivative measures how a multivariable function changes when one independent variable moves and the others are treated as constants. The selected variable matters, so fₓ and fᵧ can describe different rates of change at the same point.

Higher partials repeat the process. Mixed partials use more than one variable in a stated order, while the gradient collects all first partial derivatives into one vector. Those ideas are related, but they answer different questions.

How to use the partial derivative calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Partial derivative with respect to x

The definition changes x by a small amount while y and any other independent variables stay fixed.

\frac{\partial f}{\partial x} = h \to 0 lim \frac{f ( x + h , y ) - f ( x , y )}{h}

A reliable way to work through it

Name the differentiation variable

Write whether you need fₓ, fᵧ, a higher partial, or a mixed variable sequence before starting.

Hold other inputs constant

Apply ordinary differentiation rules to the selected variable and treat the remaining independent variables as constants.

Respect the order

For mixed partials, differentiate in the written order and do not swap it unless the needed continuity conditions hold.

Worked example

Find ∂f/∂x for f(x,y) = x²y + sin(xy)

\frac{\partial}{\partial x} (x^{2} y) = 2 x y

Treat y as a constant while differentiating x².

\frac{\partial}{\partial x} sin (x y) = y cos (x y)

Use the chain rule; the x derivative of xy is y.

f_{x} = 2 x y + y cos (x y)

Add the two partial derivatives.

The partial derivative with respect to x is 2xy + y cos(xy).

Common mistakes to check

Differentiating every variable

Only the selected variable changes. Other independent variables act like constants in that partial derivative.

Leaving the variable unstated

A multivariable function can have several different first partial derivatives.

Confusing partial and total change

A partial derivative holds other inputs fixed; a total derivative along a path includes how those inputs change.

Derivative CalculatorDifferentiate a function, identify each rule, and simplify the result one step at a time.Implicit Differentiation CalculatorDifferentiate an equation where y is not isolated, collect the y′ terms, and solve for dy/dx.Limit CalculatorStudy what a function approaches from the left, right, or at infinity, even when direct substitution fails.Taylor Series CalculatorBuild a Taylor or Maclaurin polynomial from a function, expansion center, and chosen degree.

Questions students ask

How is a partial derivative different from an ordinary derivative?

It changes one independent variable while holding the other independent variables fixed.

What is a mixed partial derivative?

It differentiates in sequence with respect to different variables, such as first x and then y.

What is the gradient?

The gradient is the vector of first partial derivatives. Where the function is differentiable, it points toward the steepest local increase.

Do partial derivatives guarantee differentiability?

No. A function can have partial derivatives at a point and still fail to be differentiable there.