Gradient Calculator

Gradient Calculator 

Find the gradient of any function existing in 2D or 3D using this calculator.

How to use this calculator?

Follow the listed steps below to use the gradient calculator.

1. Choose the dimension type i.e. two points or three points. 
2. Enter the coordinates.
3. Click Calculate.

What is the Gradient of a function?

The gradient of a function provides a measure of how that function changes in different directions. It's particularly important in the context of multivariable calculus and for functions of several variables.

For a Function of One Variable:

For a function f(x) of a single variable, the derivative f’(x) gives the slope or rate of change of the function at any point x. This is analogous to the gradient for a function of a single variable.

For a Function of Multiple Variables:

For a function f(x,y,z,...) of multiple variables, the gradient is a vector that points in the direction of the steepest ascent of the function. Its magnitude gives the rate of maximum increase of the function. 

Mathematically, the gradient of a function f in two-dimensional space (for example) is given by:

𝛁f = (∂f/∂x, ∂f/∂y)

Where ∂f/∂x and ∂f/dy are partial derivates of f with respect to x and y respectively. 

If f is a function in three-dimensional space, the gradient is:

𝛁f = (∂f/∂x, ∂f/dy, ∂f/∂z)

How to find the gradient of a function?

Let's start with a function of two variables and find its gradient.

Example:

Find the gradient of the function:

f(x,y)=3x^2y − 4y^3

Solution:

The gradient, ∇f is given by: ∇f=( ∂f/∂x, ∂f/∂y)

Find ∂f/∂x :

Differentiate f(x,y) with respect to x, treating y as a constant.

∂/∂x (3x^2y) = 6xy

∂/∂y (−4y^3) = 0

So,

∂f/∂x = 6xy

Find ∂f/∂y:

Differentiate f(x,y) with respect to y, treating x as a constant.

∂/∂y (3x^2y) = 3x^2

∂/∂y (−4y^3) = −12y^2

So,

∂f/∂y = 3x^2 −12y^2 

Combining the results:

∇f = (6xy, 3x^2 −12y^2)

So, the gradient of the function f(x,y) = 3x^2y − 4y^3 is:

∇f = (6xy, 3x^2 −12y^2)

Now, if you want the gradient at a particular point, say (x,y) = (1,2), simply plug in those values into the gradient to get the specific vector at that point.

Applications:

1. The concept of the gradient is fundamental in optimization problems, physics (like in the context of electric and gravitational fields), machine learning (especially in optimization algorithms like gradient descent), and more.

2. In essence, the gradient provides information about the direction and rate of the steepest increase of a function. If you're trying to find the minimum or maximum of a function, the gradient will be of central importance.