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Gradient Calculator

Select points, enter the function, and point values to calculate the gradient of the line using this gradient calculator, with the steps displayed.

Function (x , y)

Function (x , y , z)

for f(x, y):

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Online Gradient Calculator

An online gradient calculator helps you calculate the gradient (slope) of a straight line using two or three points. It can also differentiate multivariable functions to determine the gradient vector with step-by-step solutions. Below, you’ll learn what a gradient is, the formulas used, and practical examples.

What is a Gradient?

In vector calculus, the gradient is the derivative of a function with multiple inputs and a single output. The gradient vector contains all partial derivatives of the function and points in the direction of the steepest increase.

In simpler terms, the gradient (or slope) measures how steep a line is:

A line rising from left to right has a positive gradient.
A line falling from left to right has a negative gradient.
A vertical line has an undefined (indeterminate) gradient.

For basic 2D problems, you can calculate the slope between two points using the formula below.

Gradient Notation

The gradient of a function f at point x is commonly written as:

∇f(x)
grad f
∂f/∂x (partial derivative)
∂_if (component form)

The directional derivative relationship is:

(∇f(a)) · v = D_vf(a)

This means the dot product of the gradient and vector v gives the rate of change of the function in the direction of v.

Gradient in Cartesian Coordinates

In three-dimensional Cartesian coordinates, the gradient of a function f(x, y, z) is:

∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

Where i, j, k are unit vectors along the x, y, and z axes.

Gradient Formula (Slope Between Two Points)

To calculate the slope of a line passing through two points:

m = (y₂ − y₁) / (x₂ − x₁)

Example

Point 1: (8, 4)
Point 2: (13, 19)

m = (19 − 4) / (13 − 8)

m = 15 / 5 = 3

The gradient of the line is 3.

Example: Gradient of a Function

Find the gradient of f(x, y) = x² + y³ at point (1, 3).

∇f = (∂f/∂x, ∂f/∂y)

∂f/∂x = 2x
∂f/∂y = 3y²

∇f(1,3) = (2(1), 3(3²))
∇f(1,3) = (2, 27)

The gradient vector at (1,3) is (2, 27).

How the Gradient Calculator Works

Input

Select coordinate dimensions (2D or 3D).
Enter a function with two or three variables.
Provide coordinate values.
Click Calculate.

Output

The calculator displays the gradient vector using ∇ notation.
Step-by-step differentiation is shown for clarity.

Frequently Asked Questions

What is a vector field gradient?

The gradient of a scalar function forms a vector field. It describes how the function changes in all directions and is often associated with conservative vector fields.

Does the gradient of a vector exist?

Yes. The gradient of a vector field is a tensor that represents how the vector changes in different directions. It is commonly expressed as a matrix of partial derivatives.

What is the gradient of a scalar function?

It is a vector pointing in the direction of maximum increase of the function. Its magnitude equals the maximum rate of change at that point.

Is the gradient a row or column vector?

The gradient is a vector. It may be written as either a row or column vector depending on context, but its meaning remains the same.

What does the ∇ symbol represent?

The symbol ∇ (nabla) represents the gradient operator. In conservative vector fields, the curl is zero: ∇ × F = 0.

Conclusion

The online gradient calculator allows you to quickly compute slopes and gradient vectors for lines and multivariable functions. It eliminates manual errors and provides accurate, step-by-step solutions for students and professionals alike.

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