Gradient Calculator

Published on: July 21, 2026

Final Answer: Free • Full Steps: Pro

This Gradient Calculator collects the partial derivatives of a function into a single vector, and shows the working line by line. Each component measures how fast the function changes along one axis while the others are held still. Together they point in the direction of steepest increase, and their length is how steep that is. Following the same pattern every time keeps each variable differentiated on its own.

Step-by-step method

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

Formula:

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

See Example 1 Hide Example 1

Problem

\(f = x y + x z + y z\)

Approach

Step-by-step method

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

Step 1

Step 1 - Partial derivative with respect to x.

In this problem:

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

Step 2

Step 2 - Set up the terms in the sum.

In this problem:

\(f\left(x\right) = x y + y z + x z\)

Step 3

Step 3 - Write the sum rule formula.

In this problem:

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

Step 4

Step 4 - Split the derivative across the addition signs.

In this problem:

\(\frac{d}{dx}\left(x y + y z + x z\right) = \frac{d}{dx}\left(x y\right) + \frac{d}{dx}\left(y z\right) + \frac{d}{dx}\left(x z\right)\)

Step 5

Step 5 - Differentiate the separated term \(x y\).

In this problem:

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

Problem

\(f\left(x\right) = x y\)

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 \(y\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

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

Step 2

Step 2 - Apply the constant multiple rule first.

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

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

Problem

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

Approach

Step-by-step method

Set up the problem.
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 \(x^{1} y\). The constant coefficient is \(y\), and the variable part is \(x^{1}\).

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

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.

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

Step 3

Step 3 - Apply the constant multiple rule.

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

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

Final Answer

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

Step 3

Step 3 - Write the power rule formula.

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

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

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Use the rewritten power form from Step 1. Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(y\) outside.

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

Step 5

Step 5 - Solve and simplify.

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

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

Final Answer

\(y\)

Step 6

Step 6 - Differentiate the separated term \(y z\).

In this problem:

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

Problem

\(f\left(x\right) = y z\)

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 \(y z\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = y z\)

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 \(y z\) is constant, its derivative is \(0\).

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

Final Answer

\(0\)

Step 7

Step 7 - Differentiate the separated term \(x z\).

In this problem:

\(\frac{d}{dx}\left(x z\right) = z\)

Problem

\(f\left(x\right) = x z\)

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 \(z\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(x\right) = zx \Leftrightarrow z{x}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

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

\(\frac{d}{dx}z{x}^{1} = z\cdot \frac{d}{dx}{x}^{1}\)

Problem

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

Approach

Step-by-step method

Set up the problem.
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 \(x^{1} z\). The constant coefficient is \(z\), and the variable part is \(x^{1}\).

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

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.

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

Step 3

Step 3 - Apply the constant multiple rule.

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

\(\frac{d}{dx}\left(x^{1} z\right) = z\cdot \frac{d}{dx}\left(x^{1}\right)\)

Final Answer

\(z\cdot \frac{d}{dx}\left(x^{1}\right)\)

Step 3

Step 3 - Write the power rule formula.

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

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

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Use the rewritten power form from Step 1. Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(z\) outside.

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

Step 5

Step 5 - Solve and simplify.

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

\(z\cdot {x}^{0} = z\cdot 1 = z\)

Final Answer

\(z\)

Step 8

Step 8 - Combine and simplify.

In this problem:

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

Step 9

Step 9 - Partial derivative with respect to y.

In this problem:

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

Step 10

Step 10 - Set up the terms in the sum.

In this problem:

\(f\left(y\right) = x y + y z + x z\)

Step 11

Step 11 - Write the sum rule formula.

In this problem:

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

Step 12

Step 12 - Split the derivative across the addition signs.

In this problem:

\(\frac{d}{dy}\left(x y + y z + x z\right) = \frac{d}{dy}\left(x y\right) + \frac{d}{dy}\left(y z\right) + \frac{d}{dy}\left(x z\right)\)

Step 13

Step 13 - Differentiate the separated term \(x y\).

In this problem:

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

Problem

\(f\left(y\right) = x y\)

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 \(x\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

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

Step 2

Step 2 - Apply the constant multiple rule first.

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

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

Problem

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

Approach

Step-by-step method

Set up the problem.
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 \(x y^{1}\). The constant coefficient is \(x\), and the variable part is \(y^{1}\).

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

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 \(x\) outside the derivative and leave \(y^{1}\) inside the derivative.

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

Final Answer

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

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: Use the rewritten power form from Step 1. Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(x\) outside.

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

Step 5

Step 5 - Solve and simplify.

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

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

Final Answer

\(x\)

Step 14

Step 14 - Differentiate the separated term \(y z\).

In this problem:

\(\frac{d}{dy}\left(y z\right) = z\)

Problem

\(f\left(y\right) = y z\)

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 \(z\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(y\right) = zy \Leftrightarrow z{y}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

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

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

Problem

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

Approach

Step-by-step method

Set up the problem.
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 \(y^{1} z\). The constant coefficient is \(z\), and the variable part is \(y^{1}\).

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

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 \(z\) outside the derivative and leave \(y^{1}\) inside the derivative.

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

Final Answer

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

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: Use the rewritten power form from Step 1. Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(z\) outside.

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

Step 5

Step 5 - Solve and simplify.

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

\(z\cdot {y}^{0} = z\cdot 1 = z\)

Final Answer

\(z\)

Step 15

Step 15 - Differentiate the separated term \(x z\).

In this problem:

\(\frac{d}{dy}\left(x z\right) = 0\)

Problem

\(f\left(y\right) = x z\)

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 \(x z\). This expression does not contain \(y\), so it is a constant.

\(f\left(y\right) = x z\)

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 \(x z\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x z\right) = 0\)

Final Answer

\(0\)

Step 16

Step 16 - Combine and simplify.

In this problem:

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

Step 17

Step 17 - Partial derivative with respect to z.

In this problem:

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

Step 18

Step 18 - Set up the terms in the sum.

In this problem:

\(f\left(z\right) = x y + y z + x z\)

Step 19

Step 19 - Write the sum rule formula.

In this problem:

\(\frac{d}{dz}\left(f\left(z\right) + g\left(z\right)\right) = \frac{d}{dz}f\left(z\right) + \frac{d}{dz}g\left(z\right)\)

Step 20

Step 20 - Split the derivative across the addition signs.

In this problem:

\(\frac{d}{dz}\left(x y + y z + x z\right) = \frac{d}{dz}\left(x y\right) + \frac{d}{dz}\left(y z\right) + \frac{d}{dz}\left(x z\right)\)

Step 21

Step 21 - Differentiate the separated term \(x y\).

In this problem:

\(\frac{d}{dz}\left(x y\right) = 0\)

Problem

\(f\left(z\right) = x y\)

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 \(x y\). This expression does not contain \(z\), so it is a constant.

\(f\left(z\right) = x y\)

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 \(x y\) is constant, its derivative is \(0\).

\(\frac{d}{dz}\left(x y\right) = 0\)

Final Answer

\(0\)

Step 22

Step 22 - Differentiate the separated term \(y z\).

In this problem:

\(\frac{d}{dz}\left(y z\right) = y\)

Problem

\(f\left(z\right) = y z\)

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 \(y\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(z\right) = yz \Leftrightarrow y{z}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

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

\(\frac{d}{dz}y{z}^{1} = y\cdot \frac{d}{dz}{z}^{1}\)

Problem

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

Approach

Step-by-step method

Set up the problem.
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 \(y z^{1}\). The constant coefficient is \(y\), and the variable part is \(z^{1}\).

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

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.

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

Step 3

Step 3 - Apply the constant multiple rule.

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

\(\frac{d}{dz}\left(y z^{1}\right) = y\cdot \frac{d}{dz}\left(z^{1}\right)\)

Final Answer

\(y\cdot \frac{d}{dz}\left(z^{1}\right)\)

Step 3

Step 3 - Write the power rule formula.

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

\(\frac{d}{dz}{z}^{n} = n\,{z}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Use the rewritten power form from Step 1. Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(y\) outside.

\(y\cdot \frac{d}{dz}{z}^{1} = y\cdot {z}^{1 - 1}\)

Step 5

Step 5 - Solve and simplify.

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

\(y\cdot {z}^{0} = y\cdot 1 = y\)

Final Answer

\(y\)

Step 23

Step 23 - Differentiate the separated term \(x z\).

In this problem:

\(\frac{d}{dz}\left(x z\right) = x\)

Problem

\(f\left(z\right) = x z\)

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 \(x\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(z\right) = xz \Leftrightarrow x{z}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

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

\(\frac{d}{dz}x{z}^{1} = x\cdot \frac{d}{dz}{z}^{1}\)

Problem

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

Approach

Step-by-step method

Set up the problem.
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 \(x z^{1}\). The constant coefficient is \(x\), and the variable part is \(z^{1}\).

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

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.

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

Step 3

Step 3 - Apply the constant multiple rule.

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

\(\frac{d}{dz}\left(x z^{1}\right) = x\cdot \frac{d}{dz}\left(z^{1}\right)\)

Final Answer

\(x\cdot \frac{d}{dz}\left(z^{1}\right)\)

Step 3

Step 3 - Write the power rule formula.

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

\(\frac{d}{dz}{z}^{n} = n\,{z}^{n - 1}\)

Step 4

Step 4 - Substitute the exponent into the formula.

In this problem: Use the rewritten power form from Step 1. Here, \(n = 1\). Substitute this exponent into the power rule, then keep the coefficient \(x\) outside.

\(x\cdot \frac{d}{dz}{z}^{1} = x\cdot {z}^{1 - 1}\)

Step 5

Step 5 - Solve and simplify.

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

\(x\cdot {z}^{0} = x\cdot 1 = x\)

Final Answer

\(x\)

Step 24

Step 24 - Combine and simplify.

In this problem:

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

Step 25

Step 25 - Collect the partials into the gradient vector.

In this problem:

\(\nabla f = \left\langle y + z,\; x + z,\; x + y \right\rangle\)

Final Answer

\(\nabla f = \left\langle y + z,\; x + z,\; x + y \right\rangle\)

See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

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

Step 1

Step 1 - Partial derivative with respect to x.

In this problem:

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

Step 2

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

In this problem:

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

Step 3

Step 3 - Write the constant multiple rule formula.

In this problem:

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

Step 4

Step 4 - Apply the constant multiple rule.

In this problem:

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

Step 5

Step 5 - Partial derivative with respect to y.

In this problem:

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

Step 6

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

In this problem:

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

Step 7

Step 7 - Write the constant multiple rule formula.

In this problem:

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

Step 8

Step 8 - Apply the constant multiple rule.

In this problem:

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

Step 9

Step 9 - Collect the partials into the gradient vector.

In this problem:

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

Final Answer

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

Example Problems

Click one to load it into the box and solve it.