Directional Derivative Notes

Published on: September 29, 2026

This Directional Derivative Calculator finds how fast a function changes as you move in a chosen direction, and shows the working line by line. The gradient is built first, and the direction is rescaled to length one, because otherwise a longer arrow would fake a faster change. The dot product of the two is the rate. Following the same pattern every time keeps the direction normalised before it is used.

Step-by-step method

Set up the problem.
Find the gradient by taking the partial derivatives.
Turn the direction into a unit vector.
Dot the gradient with the unit vector.
Evaluate at the given point.

Formula:

\(D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}, \quad \mathbf{u} = \frac{\mathbf{v}}{\lVert \mathbf{v} \rVert}\)

See Example 1 Hide Example 1

Problem

\(f = x^{2} y,\; \mathbf{v} = \langle 3, 4 \rangle,\; P = (1, 2)\)

Approach

Step-by-step method

Set up the problem.
Find the gradient by taking the partial derivatives.
Turn the direction into a unit vector.
Dot the gradient with the unit vector.
Evaluate at the given point.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x^{2} y,\; \mathbf{v} = \langle 3, 4 \rangle,\; P = (1, 2)\)

Step 2

Step 2 - Find the gradient by taking the partial derivatives.

In this problem:

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

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.

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

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 \(x^{2}\), and the variable part is \(y\).

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

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 \(y\) inside the derivative.

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

Step 3

Step 3 - Turn the direction into a unit vector.

In this problem:

\(\mathbf{u} = \left\langle \frac{3}{5},\; \frac{4}{5} \right\rangle\)

Step 4

Step 4 - Dot the gradient with the unit vector.

In this problem:

\(D_{\mathbf{u}} f = \frac{2 x \left(2 x + 3 y\right)}{5}\)

Step 5

Step 5 - Evaluate at the given point.

In this problem:

\(D_{\mathbf{u}} f(1, 2) = \frac{16}{5}\)

Final Answer

\(D_{\mathbf{u}} f = \frac{16}{5}\)

See Example 2 Hide Example 2

Problem

\(f = x^{2} + y^{2},\; \mathbf{v} = \langle 1, 1 \rangle,\; P = (1, 2)\)

Approach

Step-by-step method

Set up the problem.
Find the gradient by taking the partial derivatives.
Turn the direction into a unit vector.
Dot the gradient with the unit vector.
Evaluate at the given point.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x^{2} + y^{2},\; \mathbf{v} = \langle 1, 1 \rangle,\; P = (1, 2)\)

Step 2

Step 2 - Find the gradient by taking the partial derivatives.

In this problem:

\(\nabla f = \left\langle 2 x,\; 2 y \right\rangle\)

Step 1

Step 1 - Set up the terms in the sum.

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

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

Step 2

Step 2 - 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 3

Step 3 - Split the derivative across the addition signs.

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

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

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

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

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

Problem

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

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 \(x^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

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

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.

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

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

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

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 \(2{x}^{1}\).

\(2{x}^{2 - 1} = 2{x}^{1} = 2 x\)

Final Answer

\(2 x\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

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

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Problem

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

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 \(y^{2}\). This expression does not contain \(x\), so it is a constant.

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

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 \(y^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dx}\left(y^{2}\right) = 0\)

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

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

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

Step 1

Step 1 - Set up the terms in the sum.

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

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

Step 2

Step 2 - 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 3

Step 3 - Split the derivative across the addition signs.

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

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

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

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

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Problem

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

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 \(x^{2}\). This expression does not contain \(y\), so it is a constant.

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

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 \(x^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(y^{2}\).

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

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

Problem

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

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 \(y^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

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

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, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

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

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 \(2{y}^{1}\).

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

Final Answer

\(2 y\)

Step 5

Step 5 - Combine and simplify.

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

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

Step 3

Step 3 - Turn the direction into a unit vector.

In this problem:

\(\mathbf{u} = \left\langle \frac{\sqrt{2}}{2},\; \frac{\sqrt{2}}{2} \right\rangle\)

Step 4

Step 4 - Dot the gradient with the unit vector.

In this problem:

\(D_{\mathbf{u}} f = \sqrt{2} \left(x + y\right)\)

Step 5

Step 5 - Evaluate at the given point.

In this problem:

\(D_{\mathbf{u}} f(1, 2) = 3 \sqrt{2} \approx 4.24\)

Final Answer

\(D_{\mathbf{u}} f = 3 \sqrt{2} \approx 4.24\)

See Example 3 Hide Example 3

Problem

\(f = x y^{2},\; \mathbf{v} = \langle 3, 4 \rangle,\; P = (2, 1)\)

Approach

Step-by-step method

Set up the problem.
Find the gradient by taking the partial derivatives.
Turn the direction into a unit vector.
Dot the gradient with the unit vector.
Evaluate at the given point.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x y^{2},\; \mathbf{v} = \langle 3, 4 \rangle,\; P = (2, 1)\)

Step 2

Step 2 - Find the gradient by taking the partial derivatives.

In this problem:

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

Step 1

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

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

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

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 \(y^{2}\) outside the derivative and leave \(x\) inside the derivative.

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

Step 1

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

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

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

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.

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

Step 3

Step 3 - Apply the constant multiple rule.

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

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

Step 3

Step 3 - Turn the direction into a unit vector.

In this problem:

\(\mathbf{u} = \left\langle \frac{3}{5},\; \frac{4}{5} \right\rangle\)

Step 4

Step 4 - Dot the gradient with the unit vector.

In this problem:

\(D_{\mathbf{u}} f = \frac{y \left(8 x + 3 y\right)}{5}\)

Step 5

Step 5 - Evaluate at the given point.

In this problem:

\(D_{\mathbf{u}} f(2, 1) = \frac{19}{5}\)

Final Answer

\(D_{\mathbf{u}} f = \frac{19}{5}\)

See Example 4 Hide Example 4

Problem

\(f = x^{2} + 3 y,\; \mathbf{v} = \langle 1, 2 \rangle,\; P = (1, 1)\)

Approach

Step-by-step method

Set up the problem.
Find the gradient by taking the partial derivatives.
Turn the direction into a unit vector.
Dot the gradient with the unit vector.
Evaluate at the given point.

Step 1

Step 1 - Set up the problem.

In this problem:

\(f = x^{2} + 3 y,\; \mathbf{v} = \langle 1, 2 \rangle,\; P = (1, 1)\)

Step 2

Step 2 - Find the gradient by taking the partial derivatives.

In this problem:

\(\nabla f = \left\langle 2 x,\; 3 \right\rangle\)

Step 1

Step 1 - Set up the terms in the sum.

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

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

Step 2

Step 2 - 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 3

Step 3 - Split the derivative across the addition signs.

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

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

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

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

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

Problem

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

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 \(x^{2}\). This is a power of \(x\), so the exponent is \(n = 2\).

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

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.

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

Step 3

Step 3 - Substitute the given power into the formula.

In this problem: Here, \(n = 2\). Substitute this exponent into the power rule, but do not simplify yet.

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

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 \(2{x}^{1}\).

\(2{x}^{2 - 1} = 2{x}^{1} = 2 x\)

Final Answer

\(2 x\)

Step 4b

Step 4b - Differentiate the separated term \(3 y\).

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

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

Problem

\(f\left(x\right) = 3 y\)

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 \(3 y\). This expression does not contain \(x\), so it is a constant.

\(f\left(x\right) = 3 y\)

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 \(3 y\) is constant, its derivative is \(0\).

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

Final Answer

\(0\)

Step 5

Step 5 - Combine and simplify.

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

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

Step 1

Step 1 - Set up the terms in the sum.

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

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

Step 2

Step 2 - 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 3

Step 3 - Split the derivative across the addition signs.

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

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

Step 4a

Step 4a - Differentiate the separated term \(x^{2}\).

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

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Problem

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

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 \(x^{2}\). This expression does not contain \(y\), so it is a constant.

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

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 \(x^{2}\) is constant, its derivative is \(0\).

\(\frac{d}{dy}\left(x^{2}\right) = 0\)

Final Answer

\(0\)

Step 4b

Step 4b - Differentiate the separated term \(3 y\).

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

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

Problem

\(f\left(y\right) = 3 y\)

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 \(3\) times \(x\). Rewrite the variable part as \(x^{1}\), so the power rule can be used with exponent \(n = 1\).

\(f\left(y\right) = 3y \Leftrightarrow 3{y}^{1}\)

Step 2

Step 2 - Apply the constant multiple rule first.

In this problem: Before using the power rule, apply the constant multiple rule because \(3\) is a constant coefficient.

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

Problem

\(f\left(y\right) = 3 y^{1}\)

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 \(3 y^{1}\). The constant coefficient is \(3\), and the variable part is \(y^{1}\).

\(3 y^{1} = 3\cdot \left(y^{1}\right)\)

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 \(3\) outside the derivative and leave \(y^{1}\) inside the derivative.

\(\frac{d}{dy}\left(3 y^{1}\right) = 3\cdot \frac{d}{dy}\left(y^{1}\right)\)

Final Answer

\(3\cdot \frac{d}{dy}\left(y^{1}\right)\)

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 \(3\) outside.

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

Step 5

Step 5 - Solve and simplify.

In this problem: Apply the power rule to \(y\), multiply by the constant \(3\), and keep the solving line in power form. This gives \(3\).

\(3\cdot {y}^{0} = 3\cdot 1 = 3\)

Final Answer

\(3\)

Step 5

Step 5 - Combine and simplify.

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

\(f'\left(y\right) = 0 + 3 = 3\)

Step 3

Step 3 - Turn the direction into a unit vector.

In this problem:

\(\mathbf{u} = \left\langle \frac{\sqrt{5}}{5},\; \frac{2 \sqrt{5}}{5} \right\rangle\)

Step 4

Step 4 - Dot the gradient with the unit vector.

In this problem:

\(D_{\mathbf{u}} f = \frac{2 \sqrt{5} \left(x + 3\right)}{5}\)

Step 5

Step 5 - Evaluate at the given point.

In this problem:

\(D_{\mathbf{u}} f(1, 1) = \frac{8 \sqrt{5}}{5} \approx 3.58\)

Final Answer

\(D_{\mathbf{u}} f = \frac{8 \sqrt{5}}{5} \approx 3.58\)