Directional Derivative Calculator

Directional Derivative Calculator

Directional derivative calculator is a helpful tool for determining the directional derivative of a function with the help of vectors. This calculator gives step-by-step solutions to the given problems.

What is a Directional derivative?

A directional derivative is a concept in multivariable calculus that provides a way to measure the rate of change of a scalar field along a specified direction in a higher dimensional space.

It generalizes the concept of partial derivatives and is defined as the derivative of the scalar field in a particular direction. 

It can be used to study the behavior of a scalar field in a particular direction, for example, in determining the maximum or minimum of a function along a given direction, or in optimizing a function subject to certain constraints.

Formula

Given a scalar field f (x1, x2, ..., xn) and a unit vector u, the directional derivative of f in the direction of u at a point P is given by the dot product of the gradient of f at P, and the vector u.

Df (P, u) = grad f(P) ·u

where grad f(P) is the gradient of f at P, a vector that points in the direction of the greatest rate of increase of f at that point.

How to calculate directional derivative? 

In the below example, the method of finding a directional derivative is explained briefly. 

Example

Find the directional derivative of the function sin(x)+cos(y) at point (2, pi/2,3) in the direction of the vector (1,2,3)

Solution

Step 1: The gradient of the function at a point

∇(sin(x) + cos(y)) _(1,2,3) = (cos(2),-1,0).

Step 2: Norm of a vector:

|u ⃗|=√(1)²+(2)²+(3)²= √14

Step 3: Normalize the vector, and divide each component with the norm.

[√14/14, √14/7,3√14/14]

Step 4: Finally, the directional derivative is the dot product of the gradient and normalized vector.

Df (P, u) = (cos (2), -1,0). (√14/14, √14/7,3√14/14)

Df (P, u) =-0.646