Partial Derivatives Calculator

Published on: July 21, 2026

Final Answer: Free • Full Steps: Pro

This Partial Derivatives Calculator differentiates a function of several variables, one variable at a time. To find a partial derivative it holds the other variables constant and then applies the ordinary derivative rules - power, product, quotient, chain, trigonometric, logarithmic, or exponential - showing every step the same way the Calculus 1 rule calculators do.

Step-by-step method

Set up the function of several variables.
For each variable, hold the other variables constant.
Differentiate with respect to that one variable using the ordinary derivative rules.
Write each partial derivative.

Formula:

\(\frac{\partial f}{\partial x} = f_{x}, \quad \frac{\partial f}{\partial y} = f_{y}\)

Example 1: partial derivatives of \(x^2*y + y^3\).

\(f = x^{2} y + y^{3}\)

Step 1 - Partial derivative with respect to x.

In this problem: Differentiate with respect to x. Treat y as constant.

\(\frac{\partial f}{\partial x}\)

Step 2 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2} y\), \(y^{3}\).

\(f\left(x\right) = x^{2} y + y^{3}\)

Step 3 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dx}\left(f\left(x\right) + g\left(x\right)\right) = \frac{d}{dx}f\left(x\right) + \frac{d}{dx}g\left(x\right)\)

Step 4 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dx}\left(x^{2} y + y^{3}\right) = \frac{d}{dx}\left(x^{2} y\right) + \frac{d}{dx}\left(y^{3}\right)\)

Step 5 - Differentiate the separated term \(x^{2} y\).

In this problem: The separated term is \(x^{2} y\). Use the power rule to get its derivative.

\(\frac{d}{dx}\left(x^{2} y\right) = 2 x y\)

Problem

\(f\left(x\right) = y{x}^{2}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2} y\). This is a constant multiple of a power of \(x\). The coefficient is \(y\), the power part is \(x^{2}\), and the exponent is \(n = 2\).

\(f\left(x\right) = y{x}^{2}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(y\) is a constant coefficient.

\(\frac{d}{dx}y{x}^{2} = y\cdot \frac{d}{dx}{x}^{2}\)

Problem

\(f\left(x\right) = x^{2} y\)

Approach

Step-by-step method

Set up the coefficient and the variable part.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x^{2} y\). The constant coefficient is \(y\), and the variable part is \(x^{2}\).

\(x^{2} y = y\cdot \left(x^{2}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(y\) outside the derivative and leave \(x^{2}\) inside the derivative.

\(\frac{d}{dx}\left(x^{2} y\right) = y\cdot \frac{d}{dx}\left(x^{2}\right)\)

Final Answer

\(y\cdot \frac{d}{dx}\left(x^{2}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dx}{x}^{n} = n\,{x}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, then keep the coefficient \(y\) outside.

\(y\cdot \frac{d}{dx}{x}^{2} = y\cdot 2{x}^{2 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(x^{2}\), multiply by the constant \(y\), and keep the solving line in power form. This gives \(2 y{x}^{1}\).

\(y\cdot 2{x}^{1} = 2 y{x}^{1} = 2 x y\)

Final Answer

\(2 x y\)

Step 6 - Differentiate the separated term \(y^{3}\).

In this problem: The separated term is \(y^{3}\). Use the constant rule to get its derivative.

\(\frac{d}{dx}\left(y^{3}\right) = 0\)

Problem

\(f\left(x\right) = y^{3}\)

Approach

Step-by-step method

Set up the problem.
Write the constant rule formula.
Apply the constant rule.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{3}\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = y^{3}\)

Step 2

Step 2 - Write the constant rule formula.

In this problem: The constant rule says the derivative of any constant is \(0\).

\(\frac{d}{dx}c = 0\)

Step 3

Step 3 - Apply the constant rule.

In this problem: Since \(y^{3}\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(y^{3}\right) = 0\)

Final Answer

\(0\)

Step 7 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(x\right) = 2 x y + 0 = 2 x y\)

Step 8 - Partial derivative with respect to y.

In this problem: Differentiate with respect to y. Treat x as constant.

\(\frac{\partial f}{\partial y}\)

Step 9 - Set up the terms in the sum.

In this problem: We are given a sum. The separate terms are \(x^{2} y\), \(y^{3}\).

\(f\left(y\right) = x^{2} y + y^{3}\)

Step 10 - Write the sum rule formula.

In this problem: The sum rule says the derivative of a sum is the sum of the derivatives.

\(\frac{d}{dy}\left(f\left(y\right) + g\left(y\right)\right) = \frac{d}{dy}f\left(y\right) + \frac{d}{dy}g\left(y\right)\)

Step 11 - Split the derivative across the addition signs.

In this problem: Apply the derivative to each separated term.

\(\frac{d}{dy}\left(x^{2} y + y^{3}\right) = \frac{d}{dy}\left(x^{2} y\right) + \frac{d}{dy}\left(y^{3}\right)\)

Step 12 - Differentiate the separated term \(x^{2} y\).

In this problem: The separated term is \(x^{2} y\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(x^{2} y\right) = x^{2}\)

Problem

\(f\left(y\right) = x^{2}{y}^{1}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(x^{2} y\). This is a constant multiple of a power of \(x\). The coefficient is \(x^{2}\), the power part is \(y\), and the exponent is \(n = 1\).

\(f\left(y\right) = x^{2}{y}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(x^{2}\) is a constant coefficient.

\(\frac{d}{dy}x^{2}{y}^{1} = x^{2}\cdot \frac{d}{dy}{y}^{1}\)

Problem

\(f\left(y\right) = x^{2} y^{1}\)

Approach

Step-by-step method

Set up the coefficient and the variable part.
Write the constant multiple rule formula.
Apply the constant multiple rule.

Step 1

Step 1 - Set up the coefficient and the variable part.

In this problem: We are given \(x^{2} y^{1}\). The constant coefficient is \(x^{2}\), and the variable part is \(y^{1}\).

\(x^{2} y^{1} = x^{2}\cdot \left(y^{1}\right)\)

Step 2

Step 2 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dy}\left(c\cdot f\left(y\right)\right) = c\cdot \frac{d}{dy}f\left(y\right)\)

Step 3

Step 3 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(x^{2}\) outside the derivative and leave \(y^{1}\) inside the derivative.

\(\frac{d}{dy}\left(x^{2} y^{1}\right) = x^{2}\cdot \frac{d}{dy}\left(y^{1}\right)\)

Final Answer

\(x^{2}\cdot \frac{d}{dy}\left(y^{1}\right)\)

Step 3

Step 3 - Write the power rule formula.

In this problem: Now use the power rule on the remaining power of \(x\).

\(\frac{d}{dy}{y}^{n} = n\,{y}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(x^{2}\) outside.

\(x^{2}\cdot \frac{d}{dy}{y}^{1} = x^{2}\cdot {y}^{1 - 1}\)

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(y\), multiply by the constant \(x^{2}\), and keep the solving line in power form. This gives \(x^{2}\).

\(x^{2}\cdot {y}^{0} = x^{2}\cdot 1 = x^{2}\)

Final Answer

\(x^{2}\)

Step 13 - Differentiate the separated term \(y^{3}\).

In this problem: The separated term is \(y^{3}\). Use the power rule to get its derivative.

\(\frac{d}{dy}\left(y^{3}\right) = 3 y^{2}\)

Problem

\(f\left(y\right) = {y}^{3}\)

Approach

Step-by-step method

Set up the problem.
If a constant coefficient is present, apply the constant multiple rule first.
Write the power rule formula.
Substitute the exponent into the power rule.
Solve and simplify.

Step 1

Step 1 - Set up the problem.

In this problem: We are given \(y^{3}\). This is a power of \(x\), so the exponent is \(n = 3\).

\(f\left(y\right) = {y}^{3}\)

Step 2

Step 2 - Write the power rule formula.

In this problem: The power rule says to move the exponent to the front, then subtract 1 from the exponent.

\(\frac{d}{dy}{y}^{n} = n\,{y}^{n - 1}\)

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 3\). Substitute this exponent into the power rule, but do not simplify yet.

\(\frac{d}{dy}{y}^{3} = 3{y}^{3 - 1}\)

Step 4

Step 4 - Solve and simplify.

In this problem: Now subtract 1 from the exponent and keep the result in power form. This gives \(3{y}^{2}\).

\(3{y}^{3 - 1} = 3{y}^{2}\)

Final Answer

\(3 y^{2}\)

Step 14 - Combine and simplify.

In this problem: Combine the derivative from each separated term, then simplify the result.

\(f'\left(y\right) = x^{2} + 3 y^{2}\)

Final answer:

\(\frac{\partial f}{\partial x} = 2 x y,\quad \frac{\partial f}{\partial y} = x^{2} + 3 y^{2}\)

Example 2: partial derivatives of \(x*y*z^2\).

\(f = x y z^{2}\)

Step 1 - Partial derivative with respect to x.

In this problem: Differentiate with respect to x. Treat y, z as constants.

\(\frac{\partial f}{\partial x}\)

Step 2 - Set up the coefficient and the variable part.

In this problem: We are given \(x y z^{2}\). The constant coefficient is \(y z^{2}\), and the variable part is \(x\).

\(x y z^{2} = y z^{2}\cdot \left(x\right)\)

Step 3 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dx}\left(c\cdot f\left(x\right)\right) = c\cdot \frac{d}{dx}f\left(x\right)\)

Step 4 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(y z^{2}\) outside the derivative and leave \(x\) inside the derivative.

\(\frac{d}{dx}\left(x y z^{2}\right) = y z^{2}\cdot \frac{d}{dx}\left(x\right)\)

Step 5 - Partial derivative with respect to y.

In this problem: Differentiate with respect to y. Treat x, z as constants.

\(\frac{\partial f}{\partial y}\)

Step 6 - Set up the coefficient and the variable part.

In this problem: We are given \(x y z^{2}\). The constant coefficient is \(x z^{2}\), and the variable part is \(y\).

\(x y z^{2} = x z^{2}\cdot \left(y\right)\)

Step 7 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dy}\left(c\cdot f\left(y\right)\right) = c\cdot \frac{d}{dy}f\left(y\right)\)

Step 8 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(x z^{2}\) outside the derivative and leave \(y\) inside the derivative.

\(\frac{d}{dy}\left(x y z^{2}\right) = x z^{2}\cdot \frac{d}{dy}\left(y\right)\)

Step 9 - Partial derivative with respect to z.

In this problem: Differentiate with respect to z. Treat x, y as constants.

\(\frac{\partial f}{\partial z}\)

Step 10 - Set up the coefficient and the variable part.

In this problem: We are given \(x y z^{2}\). The constant coefficient is \(x y\), and the variable part is \(z^{2}\).

\(x y z^{2} = x y\cdot \left(z^{2}\right)\)

Step 11 - Write the constant multiple rule formula.

In this problem: The constant multiple rule says a constant coefficient stays outside the derivative.

\(\frac{d}{dz}\left(c\cdot f\left(z\right)\right) = c\cdot \frac{d}{dz}f\left(z\right)\)

Step 12 - Apply the constant multiple rule.

In this problem: Move the constant coefficient \(x y\) outside the derivative and leave \(z^{2}\) inside the derivative.

\(\frac{d}{dz}\left(x y z^{2}\right) = x y\cdot \frac{d}{dz}\left(z^{2}\right)\)

Final answer:

\(\frac{\partial f}{\partial x} = y z^{2},\quad \frac{\partial f}{\partial y} = x z^{2},\quad \frac{\partial f}{\partial z} = 2 x y z\)