## Introduction to Gradient Calculator:

Gradient calculator is an online solver that is used to **find the gradient** of a given function with respect to its variable (x,y,z). It determines the vector function f(x1,x2,…,xn) that is solved by a partial derivatives method in two or three dimensional space.

The gradient formula calculatoris a useful tool for students,teachers and researchers because It simplifies the process of finding the gradients, especially for complex functions with multiple variables as it is used in various fields like physics, economics, engineering and more.

## What is Gradient?

Gradient gives the direction of the steepest ascent in the scalar field, however to identify points of equilibrium and understand areas of rapid change, we determine laplacian of the function.

The gradient function ∇f(x) is a vector that represent gradient with a nebula symbol. The value of the gradient function may be negative or positive as per the given function.The negative value of a vector function shows the change in direction.

## Formula of Gradient:

The gradient has a **scalar-valued function** f(x), where x=(x1,x2,…,xn) represents the vector of variables (x,y,z), the gradient function is represent as ∇f(x) in two or three dimension that is defined as:

$$ \nabla f(x,y) \;=\; f_x(x,y) \hat{i} + f_y(x,y)\hat{j} $$

$$ \nabla f(x,y,z) \;=\; f_x(x,y,z) \hat{i} + f_y(x,y,z) \hat{j} + f_z(x,y,z) \hat{k} $$

## How to Calculate the Gradient?

To **calculate the gradient** function we use the first order of partial derivation on the given function with respect to each of its variables one by one. Let's look at the method of finding gradient of a function in steps which is:

**Step 1:**

First identify the given function f(x,y,z) that gradient you want to evaluate.

**Step 2:**

After identification, partial differential the function f(x,y,z) with respect to each of its variables (x,y,z). It must be remembered while doing partial derivation only the respect variable function is different and other function values are treated as constant.

**Step 3:**

Add the result of partial derivatives into a vector of scalar function into the gradient vector formula ∇f(x) that is given as:

$$ \nabla f(x,y) \;=\; f_x(x,y) \hat{i} + f_y(x,y)\hat{j} $$

$$ \nabla f(x,y,z) \;=\; f_x(x,y,z) \hat{i} + f_y(x,y,z) \hat{j} + f_z(x,y,z) \hat{k} $$

## Practical Example Of Gradient:

An **example of gradient** function with solution given an idea about the procedure of gradient method for different dimensional space in vector analysis.

### Example: Find the gradient ▽f(x,y) of the following,

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

**Solution**:

The given function is,

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

Differentiate the function with respect to x and y one by one,

$$ f_x(x,y) \;=\; 2x - y $$

$$ f_y(x,y) \;=\; -x + 6y $$

The gradient of two dimensional space is,

$$ \nabla f(x,y) \;=\; f_x(x,y) \hat{i} + f_y(x,y) \hat{j} $$

Put the above derivative function result in this gradient formula,

$$ =\; (2x - y) \hat{i} + (-x + 6y) \hat{j} $$

The result of given function f(x,y) is,

$$ =\; (2x - y)\hat{i} + (-x + 6y)\hat{j} $$

## How to Use the Gradient of a Line Calculator?

Gradient vector calculator has a simple design that helps you solve the given algebraic function of 2D,3D space. You just need to put your problem in it and follow some important instructions so that you get the result of the gradient function without any trouble. These instructions are:

**Enter the scalar valued**gradient function that you want to evaluate in the input fields.- Recheck your given input value to get the exact solution of the gradient function.
- The Calculate button provides you with solutions to given gradient problem in two or may be three dimensional.
- If you want to check the accuracy of our gradient function calculator, then you can use the load example and get an idea about its working procedure.
- Click the “Recalculate” button for the evaluation of more examples of the gradient function with the solution.

## Outcome of Gradient Calculator:

Gradient descent calculator provides you with a **solution** as per your input gradient function problem when you click on the calculate button. It may include as:

**In the Result Box**:

Click on the result button to get the solution of the gradient question.

**Steps Box**:

When you click on the steps option, you get the solution of gradient questions in a step-by-step process.

## Benefits of Using Gradient of a Function Calculator:

The gradient vector calculator has multiple benefits that you get whenever you use it for the calculation of gradient function to get the solution of gradient questions. These benefits are:

- Our gradient formula calculator is a
**trustworthy tool**as it always provides you with accurate solutions of given gradient function problems. - It is an efficient tool that provides solutions in a stepwise process of the given gradient problems in a few seconds.
- Gradient of a line calculator is a learning tool that helps you to easily grab the concept of gradient of a vector field very easily without going to any tutor.
- Gradient calculator is a handy tool that solves gradient problems as per the given gradient problems and you do not need to give any type of external effort during calculation
- The gradient function calculatoris a free tool that allows you to use it for the calculation of gradient problems without spending.
- The gradient of line calculatoris an easy-to-use tool, anyone, even a beginner can easily use it for the solution of gradient function problems.
- The gradient vector calculator can save you tons of time that you use in finding the solution to gradient questions as it provides you results immediately.