Directional Derivative Calculator

Calculate a directional derivative

Enter a function, a point, and a direction vector. The direction vector is automatically normalized to a unit vector.

Quick examples

How it works

For a scalar function \(f\), the directional derivative in unit direction \(\mathbf{u}\) is:

\[ D_{\mathbf{u}}f(\mathbf{a}) = \nabla f(\mathbf{a}) \cdot \mathbf{u} \]

The calculator performs four steps:

Finds each partial derivative.
Builds the gradient vector.
Normalizes your direction vector.
Computes the dot product.

Directional Derivative Formula

For a function of two variables, \(f(x,y)\), the gradient is \(\nabla f=(f_x,f_y)\). If the direction vector is \(\mathbf{v}=(a,b)\), its magnitude is \(\|\mathbf{v}\|=\sqrt{a^2+b^2}\). The corresponding unit vector is:

\[ \mathbf{u} = \frac{\mathbf{v}} {\|\mathbf{v}\|} \]

The directional derivative is then:

\[ D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u} \]

What Is a Directional Derivative?

A directional derivative measures how rapidly a multivariable function changes when moving from a specific point in a particular direction. Unlike an ordinary derivative, which describes change along one coordinate axis, a directional derivative allows the direction of movement to be chosen.

The gradient is especially important because it points in the direction of greatest increase of a differentiable function. The maximum directional derivative at a point is therefore the magnitude of the gradient.

Example

Consider:

\[ f(x,y)=x^2+y^2 \]

Its gradient is:

\[ \nabla f=(2x,2y) \]

At the point \((1,2)\):

\[ \nabla f(1,2)=(2,4) \]

Suppose the direction vector is \((1,1)\). Its unit vector is:

\[ \mathbf{u} = \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) \]

Therefore:

\[ D_{\mathbf{u}}f = (2,4) \cdot \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right) = \frac{6}{\sqrt{2}} \approx 4.24264 \]

Supported Functions

The calculator supports ordinary arithmetic and common calculus functions including:

addition, subtraction, multiplication and division
powers and exponents
implicit multiplication such as \(2x\) and \(xy\)
\(\sin(x)\), \(\cos(x)\), \(\tan(x)\)
\(\sec(x)\), \(\csc(x)\), \(\cot(x)\)
\(\arcsin(x)\), \(\arccos(x)\), \(\arctan(x)\)
\(\ln(x)\), \(\log(x)\), \(\log_{10}(x)\)
\(\exp(x)\)
\(\sqrt{x}\)
\(|x|\) using abs(x)

Related Summe.org Calculators

Gradient Calculator · Partial Derivative Calculator · Hessian Calculator · Jacobian Calculator · Critical Point Calculator · Divergence Calculator · Curl Calculator · Derivative Calculator

Frequently Asked Questions

Does the calculator normalize the direction vector?

Yes. You can enter any non-zero direction vector. The calculator automatically divides it by its magnitude to create a unit direction vector.

Can I use three variables?

Yes. Functions involving \(x\), \(y\), and \(z\) are supported. For a three-variable function, enter three coordinates for the point and three components for the direction.

What happens if I enter a zero direction vector?

A zero vector has no direction and therefore cannot be normalized. The calculator displays an error instead of attempting to calculate a meaningless directional derivative.

What is the difference between a gradient and a directional derivative?

The gradient is a vector containing all first partial derivatives. A directional derivative is a scalar obtained by taking the dot product of the gradient with a unit vector in the chosen direction.