Gradient Calculator

Published on: July 21, 2026

Final Answer: Free • Full Steps: Pro

This Gradient Calculator finds the gradient vector \(\nabla f\) of a function of several variables. It takes the partial derivative with respect to each variable - holding the others constant and applying the ordinary derivative rules step by step - then collects the partials into the gradient vector.

Step-by-step method

Set up the function of several variables.
Take the partial derivative with respect to each variable (holding the others constant).
Collect the partials into the gradient vector.

Formula:

\(\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle\)

Example 1: gradient 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.

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

Step 15 - Collect the partials into the gradient vector.

In this problem: Place each partial derivative into the gradient vector.

\(\nabla f = \left\langle 2 x y,\; x^{2} + 3 y^{2} \right\rangle\)

Final answer:

\(\nabla f = \left\langle 2 x y,\; x^{2} + 3 y^{2} \right\rangle\)

Example 2: gradient 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.

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

Step 13 - Collect the partials into the gradient vector.

In this problem: Place each partial derivative into the gradient vector.

\(\nabla f = \left\langle y z^{2},\; x z^{2},\; 2 x y z \right\rangle\)

Final answer:

\(\nabla f = \left\langle y z^{2},\; x z^{2},\; 2 x y z \right\rangle\)