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Directional Derivative Calculator

Select the coordinates' type and enter all required parameters in their respective fields. The calculator will quickly calculate the directional derivative for the entered function.

for (x , y)

for (x , y , z)

coordinate f(x, y):

U1:

U2:

x Function:

y Function:

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Our online Directional Derivative Calculator computes the directional derivative and gradient of a multivariable function at a specific point along a given vector. It also provides step-by-step solutions to help understand how the function changes in different directions.

Below, we explain how to calculate directional derivatives with formulas and examples.

What Is a Directional Derivative?

In mathematics, a directional derivative measures the rate of change of a function at a point in the direction of a given vector. It extends the concept of partial derivatives to any specified direction, showing how the function changes while holding other variables constant.

(Illustrative image)

Directional Derivative Formula

For a function \(f\) and a vector \(v\), the directional derivative of \(f\) at point \(p\) is denoted as:

\(D_v f(p)\)
\(f'_v(p)\)
\(\nabla_v f(p)\)

If \(f(x_1, x_2, \dots, x_n)\) is a scalar function and \(v = (v_1, v_2, \dots, v_n)\), the directional derivative is given by:

\(\nabla_v f(x) = \lim_{h \to 0} \frac{f(x + h v) - f(x)}{h}\)

The Gradient

The gradient, \(\nabla f\), is a vector pointing in the direction of steepest ascent. Its magnitude equals the maximum rate of change. The directional derivative along a unit vector \(u\) can be computed as a dot product:

\(D_u f = \nabla f \cdot u\)

Example of Directional Derivative

Let \(f(x, y) = 14 - x^2 - y^2\) and consider the point \(M = (3,4)\). Compute the directional derivative in the following directions:

Toward \(N = (5,6)\)
In the direction of \(\langle 2, -1 \rangle\)
Toward the origin

Solution:

First, compute partial derivatives:

\(f_x = -2x \Rightarrow f_x(3,4) = -6\)

\(f_y = -2y \Rightarrow f_y(3,4) = -8\)

Unit vector in the direction of \(N - M = (2,2)\):

\(\vec u_1 = \frac{(2,2)}{\sqrt{2^2 + 2^2}} = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\)

Directional derivative:

\(D_{\vec u_1} f(3,4) = (-6)(\frac{1}{\sqrt{2}}) + (-8)(\frac{1}{\sqrt{2}}) = \frac{-14}{\sqrt{2}} \approx -9.9\)

Hence, the rate of change at \(M=(3,4,9)\) in the direction of \(\vec u_1\) is approximately -9.9.

How the Directional Derivative Calculator Works

Input:

Enter the function \(f(x_1, x_2, …)\).
Provide the point coordinates.
Enter the direction vector or points defining the direction.
Click "Calculate" to see the results.

Output:

Directional derivative of the function along the specified vector.
Gradient vector at the given point, computed from partial derivatives.
Step-by-step solution showing all calculations.

FAQ

What is a gradient?

The gradient of a function points in the direction of steepest increase. Its magnitude gives the rate of change in that direction.

What is the difference between gradient and directional derivative?

The directional derivative gives the rate of change in a specific direction. The gradient shows the direction of maximum increase and can be used to compute directional derivatives in any direction.

Is the first-order derivative the same as the gradient?

In one variable, the first-order derivative measures the slope. The gradient generalizes this concept to multiple variables, giving slopes along all directions.

Why is the directional derivative important?

It measures the instantaneous rate of change along any vector, which is essential in optimization, physics, and multivariable modeling.

In which direction is the directional derivative largest?

It is largest in the direction of the gradient vector and smallest (most negative) in the opposite direction.

Can directional derivatives be negative?

Yes, a negative directional derivative means the function decreases in that direction.

Conclusion

The Directional Derivative Calculator extends partial derivatives to any direction, computing gradients and directional derivatives efficiently. It reduces manual errors and provides clear, step-by-step results.

References:

From Wikipedia: Directional derivative

From LibreTexts: Directional Derivatives and Gradient

From Active Calculus: Directional Derivatives and Applications

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