Gradient Calculator

Gradient Calculator 

This gradient calculator finds the partial derivatives of functions. You can enter the values of a vector line passing from 2 points and 3 points. For detailed calculation, click “show steps”.

What is a Gradient?

The gradient is similar to the slope. It is represented by ∇(nabla symbol). A gradient in calculus and algebra can be defined as:

“A differential operator applied to a vector-valued function to yield a vector whose components are the partial derivatives of the function with respect to its variables.”

Gradient Formula

$\nabla f (x, y, z) = \Huge [ \small \begin{array}{ccc} \dfrac{df}{dx}\ \ \dfrac{df}{dy}\ \ \dfrac{df}{dz} \end{array} \Huge]$

A gradient is calculated by finding the partial derivative of the function with respect to the variable. 

How to calculate gradient?

Example:

A function is x² + y². Find its gradient for point (2,2).

Solution:

Step 1: Find the differentiate.

 

$\nabla f = \Big( \dfrac{df}{dx}, \dfrac{df}{dy} \Big)=ddx[d(x2+ y2) , = ddy[d(x2+ y2)$

 

$=\dfrac{d}{dx}[d(x^2+ y^2) \enspace ,=\dfrac{d}{dy}[d(x^2+ y^2)$

 

$=d(\dfrac{d}{dx}[x^2]+\dfrac{d}{dx}[y^2]) \enspace, = d(\dfrac{d}{dy}[x^2]+\dfrac{d}{dy}[y^2])$

 

$= d (2x + 0) \enspace , = d (0 + 2y)$

 

$= 2x , = 2y$

 

Step 2: Plug in the point.

 

$\nabla f (x,y) = [(2)(x) , (2)(y)]$

 

$\nabla f (2,2) = [(2)(2) , (2)(2)]$

 

= 4 , 4

Step 3: Write equation.

 

$\nabla(x^2+y^2)(x,y)=(2x , 2y)$

 

$\nabla(x^{2} + y^{3})|_{(x,y)=(2,2)} = (4,4)$