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# Directional Derivative Calculator

To use the directional derivative calculator, enter the function in the respective box, enter the points x, y, & z, and click the calculate

## 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 the step by steps solutions to the given problems.

## What is a Directional derivative?

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 (x_{1}, x_{2}, ..., x_{n}) 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