Calculators / Calculus 3 / Partial Derivatives Calculator

Partial Derivatives Calculator

Published on: July 5, 2026

This Partial Derivatives Calculator helps you find first-order, second-order, and mixed partial derivatives of a multivariable function. Enter the function and specify the derivatives you want, such as fx, fxx, or fxy, and the calculator will differentiate with respect to the chosen variables step by step. If no requests are given, it computes the first partial derivatives for the variables it detects. It is a simple way to check answers, understand the method clearly, and practise multivariable calculus step by step.

Allowed requests: fx, fy, fz, fxx, fyy, fzz, fxy, fyx, fxz, fzx, fyz, fzy. If you don’t type requests, it will compute only the first partials for the variables it detects.

Step-by-step method

  1. Enter f(x,y) or f(x,y,z), then list what you want (fx, fxx, fxy, ...).
  2. Differentiate using substeps (chain/product rule shown clearly).
  3. Combine in its own substep.

Note:

Input format: f(x,y)=..., fx, fxx, fxy. Allowed: fx, fy, fz, fxx, fyy, fzz, fxy, fyx, fxz, fzx, fyz, fzy.

Example 1: f(x,y)=yx^2+sin(xy), fx, fxy

Step 1 - Identify the function and requests.

In this problem: Variables: x, y | Requested: fx, fxy

f(x, y) = yx2 + sin(xy)

Step 2A - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂x
(yx2) = 2xy

Step 2B - Differentiate term 2.

In this problem: Rule shown in the step breakdown.

∂
∂x
(sin(xy)) = cos(xy)·
∂
∂x
(xy)

Step 2C - Differentiate term 2.

In this problem: Rule shown in the step breakdown.

∂
∂x
(xy) = y

Step 2D - Differentiate term 2.

In this problem: Rule shown in the step breakdown.

∂
∂x
(sin(xy)) = cos(xy)·y = y·cos(xy) = y·cos(xy)

Step 2E - Combine the term results.

In this problem: Add the derivatives from the previous substeps.

fx = 2xy + y·cos(xy) = y(2x + cos(xy))

Step 3A - Set up the derivative.

In this problem: Use fx from the previous main step, then differentiate with respect to y.

fxy =
∂
∂y
(y·cos(xy) + 2xy)

Step 3B - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂y
(y·cos(xy)) =
∂
∂y
(y)·cos(xy) + y·
∂
∂y
(cos(xy))

Step 3C - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂y
(y) = 1

Step 3D - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂y
(cos(xy)) = −sin(xy)·
∂
∂y
(xy)

Step 3E - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂y
(xy) = x

Step 3F - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂y
(cos(xy)) = −sin(xy)·x = x·−sin(xy) = −x·sin(xy)

Step 3G - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂y
(y·cos(xy)) = 1·cos(xy) + y·−x·sin(xy)

Step 3H - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂y
(y·cos(xy)) = −xy·sin(xy) + cos(xy)

Step 3I - Differentiate term 2.

In this problem: Rule shown in the step breakdown.

∂
∂y
(2xy) = 2x

Step 3J - Combine the term results.

In this problem: Add the derivatives from the previous substeps.

fxy = −xy·sin(xy) + cos(xy) + 2x = 2x − xy·sin(xy) + cos(xy)

Final answer: Computed: fx, fxy

Example 2: f(x,y)=3x^3-2xy+sin(xy)+y^5, fy, fyy

Step 1 - Identify the function and requests.

In this problem: Variables: x, y | Requested: fy, fyy

f(x, y) = y5 + 3x3 − 2xy + sin(xy)

Step 2A - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂y
(y5) = 5y4

Step 2B - Differentiate term 2.

In this problem: Rule shown in the step breakdown.

∂
∂y
(3x3) = 0

Step 2C - Differentiate term 3.

In this problem: Rule shown in the step breakdown.

∂
∂y
(−2xy) = −2x

Step 2D - Differentiate term 4.

In this problem: Rule shown in the step breakdown.

∂
∂y
(sin(xy)) = cos(xy)·
∂
∂y
(xy)

Step 2E - Differentiate term 4.

In this problem: Rule shown in the step breakdown.

∂
∂y
(xy) = x

Step 2F - Differentiate term 4.

In this problem: Rule shown in the step breakdown.

∂
∂y
(sin(xy)) = cos(xy)·x = x·cos(xy) = x·cos(xy)

Step 2G - Combine the term results.

In this problem: Add the derivatives from the previous substeps.

fy = 5y4 + 0 + −2x + x·cos(xy) = −2x + 5y4 + x·cos(xy)

Step 3A - Set up the derivative.

In this problem: Use fy from the previous main step, then differentiate with respect to y.

fyy =
∂
∂y
(−2x + 5y4 + x·cos(xy))

Step 3B - Differentiate term 1.

In this problem: Rule shown in the step breakdown.

∂
∂y
(−2x) = 0

Step 3C - Differentiate term 2.

In this problem: Rule shown in the step breakdown.

∂
∂y
(5y4) = 20y3

Step 3D - Differentiate term 3.

In this problem: Rule shown in the step breakdown.

∂
∂y
(x·cos(xy)) = −x2·sin(xy)

Step 3E - Combine the term results.

In this problem: Add the derivatives from the previous substeps.

fyy = 0 + 20y3 + −x2·sin(xy) = 20y3 − x2·sin(xy)

Final answer: Computed: fy, fyy