Gradient Calculator
Compute the gradient vector of a polynomial function in x and y at a given point.
Inputs
Enter function using x and y as variables.
∂f/∂x
3.00000000
∂f/∂y
3.00000000
|∇f|
4.24264069
Step by step
Values used
f(x, y) = x^2 + y^2 + x*y; x value = 1; y value = 1
Gradient vector
∇f = (∂f/∂x, ∂f/∂y)
Magnitude
|∇f| = sqrt((∂f/∂x)² + (∂f/∂y)²)
∂f/∂x
= 3.00000000
∂f/∂y
= 3.00000000
|∇f|
= 4.24264069
How it works
The gradient ∇f = (∂f/∂x, ∂f/∂y) points in the direction of steepest ascent. Its magnitude gives the rate of maximum increase. Computed numerically using central differences for each partial derivative.
Formulas
Gradient vector
∇f = (∂f/∂x, ∂f/∂y)
- ∇f
- Gradient vector
- ∂f/∂x
- Partial derivative w.r.t. x
- ∂f/∂y
- Partial derivative w.r.t. y
Magnitude
|∇f| = sqrt((∂f/∂x)² + (∂f/∂y)²)
- |∇f|
- Gradient magnitude
Frequently Asked Questions
What does the gradient represent?
The gradient points in the direction of greatest increase of the function and its magnitude gives the rate of that increase.
How is the gradient used in optimization?
Gradient descent moves in the negative gradient direction to find function minima, which is the basis of machine learning optimization.