Directional Derivative Calculator

Published on: July 21, 2026

Final Answer: Free • Full Steps: Pro

This Directional Derivative Calculator finds how fast \(f(x, y)\) changes in a chosen direction at a point. It computes the gradient, converts the direction into a unit vector, dots them together, and evaluates - showing every step.

Step-by-step method

Set up the function, the direction vector, and the point.
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:

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

Example 1:

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

Step 1 - Set up the function, the direction vector, and the point.

In this problem: We are given \(f = x^{2} y\), direction \(\mathbf{v} = \langle 3, 4 \rangle\), and the point \((1, 2)\).

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

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

In this problem: The gradient is \(\nabla f = \langle 2 x y, x^{2} \rangle\).

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

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 - Turn the direction into a unit vector.

In this problem: The magnitude is \(\lVert \mathbf{v} \rVert = 5\), so the unit vector is \(\mathbf{u} = \langle \frac{3}{5}, \frac{4}{5} \rangle\).

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

Step 4 - Dot the gradient with the unit vector.

In this problem: Dotting the gradient with the unit vector gives \(D_{\mathbf{u}} f = \frac{2 x \left(2 x + 3 y\right)}{5}\).

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

Step 5 - Evaluate at the given point.

In this problem: Evaluating at \((1, 2)\) gives \(\frac{16}{5}\).

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

Final answer:

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

Example 2:

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

Step 1 - Set up the function, the direction vector, and the point.

In this problem: We are given \(f = x^{2} + y^{2}\), direction \(\mathbf{v} = \langle 1, 1 \rangle\), and the point \((1, 2)\).

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

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

In this problem: The gradient is \(\nabla f = \langle 2 x, 2 y \rangle\).

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

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 - Turn the direction into a unit vector.

In this problem: The magnitude is \(\lVert \mathbf{v} \rVert = \sqrt{2}\), so the unit vector is \(\mathbf{u} = \langle \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \rangle\).

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

Step 4 - Dot the gradient with the unit vector.

In this problem: Dotting the gradient with the unit vector gives \(D_{\mathbf{u}} f = \sqrt{2} \left(x + y\right)\).

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

Step 5 - Evaluate at the given point.

In this problem: Evaluating at \((1, 2)\) gives \(3 \sqrt{2} \approx 4.24\).

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

Final answer:

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