## Introduction to Directional Derivative Calculator:

Directional derivative calculator angle is the best online source that helps you to **find** the directional derivative of a given function in multidimensional space. Our tool determines the rate of a change of function at a specific point.

The directional derivative at a point calculator is a handy tool for students, teachers, professionals, or researchers who want to get a solution of multivariable function in a specific direction of scalar field.

## What is a Directional Derivative?

The directional derivative is a process that is used to find the **multivariable function** f(x, y, z,…) at a point a = (a1, a2, a3,…) to measure the rate at which the given function changes, going towards a in the direction of a vector u = (u1, u2, u3,…).

The directional derivative process is used for partial derivative functions where you can find the scalar function in a particular direction. As it determines the maximum or minimum value of several functions along the given direction that describes function behavior in a scalar field.

### Formula of Directional Derivative:

The directional derivative is mathematically defined as a multivariable function f, of a given function at point a in the direction of vector u. The directional derivative **formula** used by the directional derivative calculator with angle is given as:

$$ D_u f(a) \;=\; \nabla f(a) . u $$

- Du: the directional derivative toward the u vector
- f(a): function at a point a
- ▽: the gradient of a multivariable function
- Here, the gradient ▽ of a function f at a point a is f(a).

$$ \nabla f(a) \;=\; \left( \frac{\partial f}{\partial x_1}(a),\; \frac{\partial f}{\partial x_2}(a),\; \frac{\partial f}{\partial x_3}(a), … \right) $$

## How to Calculate Directional Derivative?

The directional derivative online calculator is an easy way to **calculate** the partial derivation in a specific direction of scalar field. The process of direction derivative calculation is given as:

**Step 1**:

Identify the given function f(x, y, z…) and a specific point value a(a1, a2,..) along the u direction.

**Step 2**:

Evaluate the gradient ▽ of a given function as it takes the partial differential with respect to its variable along the direction.

**Step 3**:

After differentiation, simplify the expression and then apply the value of point a(a1, a2,..)

**Step 4**:

After putting the point value in the differentiation function simplify it to get the solution.

## Solved Example of Directional Derivative:

The directional derivative calculator angle solves the directional derivative problem without any hurdle. So, here’s an **example** to let you know how it actually calculates,

**Example**:

Determine the directional derivative of the following,

$$ f(x,y) \;=\; x^2 - xy + 3y^2\; in\; the\; direction\; of\; \vec{u} \;=\; (cos\; \theta) \hat{i} + (sin\; \theta) \hat{j} $$

**Solution**:

Identify the given function f(x, y, z…) at a particular point and direction,

$$ f(x,y) \;=\; x^2 - xy + 3y^2 $$

$$ \vec{u} \;=\; (cos\; \theta) \hat{i} + (sin\; \theta) \hat{j} $$

Compute the gradient of the function:

$$ D_{\vec{u}} f(x, y, z) \;=\; f_x (x,y) cos\; \theta + f_y (x,y) sin\; \theta $$

$$ \;=\; (2x - y) \frac{3}{5} + (-x + 6y) \frac{4}{5} $$

$$ =\; \frac{6x}{5} - \frac{3y}{5} - \frac{4x}{5} + \frac{24y}{5} $$

$$ =\; \frac{2x + 21y}{5} $$

Put the value of given point a in the above function and simply use the expression to get the solution,

$$ D_{\vec{u}} f(1,2) \;=\; \frac{2(-1) + 21(2)}{5} \;=\; \frac{-2 + 42}{5} \;=\; 8 $$

## How to Use Directional Derivative Calculator?

The directional derivative at a point calculator has a simple design that makes it easy for you to use it for the evaluation of directional derivative problems. Follow some simple steps that are given as:

- Enter the given function in the given input field that you want to differentiate with a specific point.
- Put the value of the direction of a vector u(u1, u2,..) in the next input field.
- Put the value of the point around which the directional derivative is evaluated.
- Check the given directional derivative function value before clicking the calculate button of the directional derivative online calculator.
- Click the “
**Calculate**” button to get the result of your directional derivative problem. - If you are trying our tool for the first time then you can use the load example option to learn more about this process.
- The “Recalculate” button brings you back to the home page where you can find more solutions of directional derivative problems.

## Results from Directional Derivative at a Point Calculator:

The directional derivative calculator with angle gives you the **solution** from a given function when you give it an input. It includes as:

**Result Option**:

When you click on the result option, it gives you a solution of partial differential function in a specific point.

**Possible Steps**:

When you click on it, this option will provide you step by step solution where you get the function in the direction of u.

**Plot Option**:

It will give you a graphical representation of directional derivative function to get a better understanding of the result.

## Advantages of Directional Derivative Online Calculator:

The directional derivative at a point calculator provides tons of **advantages** and helps you to calculate the directional derivative of a function. These advantages are:

- It is a free-of-cost tool so you can use it for free to find directional derivative problems with solutions.
- It gives you conceptual clarity for directional derivative process when you use it to solve multiple examples.
- It saves the time and effort that you consume when calculating complex directional derivative functions to find the maximum or minimum value manually.
- It is an adaptable tool that provides you accurate solutions whenever you use it to calculate directional derivatives.
- Our directional derivative calculator angle enables you to use it multiple times for the evaluation of directional derivative problems.
- It is a handy tool so you can access it through an online platform from anywhere.