Gradient Vector Calculator

Calculate the gradient vector of f(x,y) = ax² + by² + cxy at any point. Shows magnitude and direction of steepest ascent.

Coefficient of x² (a)

Coefficient of y² (b)

Coefficient of xy (c)

x Point

y Point

Tab to move between fields

∂f/∂x

∂f/∂y

Magnitude |∇f|7.2111

Direction56.31°

How to Use This Calculator

Enter the coefficients a, b, and c that define f(x,y) = ax² + by² + cxy.
Enter the x and y coordinates of the point where you want to evaluate the gradient.
The calculator computes the partial derivatives ∂f/∂x = 2ax + cy and ∂f/∂y = 2by + cx at that point.
Gradient magnitude |∇f| = √((∂f/∂x)² + (∂f/∂y)²) tells you how steep the function is at that point.
Direction gives the angle (in degrees) of the gradient vector, pointing toward the steepest ascent of f.
The bar chart displays ∂f/∂x, ∂f/∂y, and |∇f| side by side to compare their magnitudes.

How the result changes with Coefficient of x² (a)

Identify Input Parameters
4 parameters
Coefficient of x² (a) = 1, Coefficient of y² (b) = 1, Coefficient of xy (c) = 0, x Point = 2 = 5 input(s) provided
Calculate ∂f/∂x
∂f/∂x
4 = 4
Calculate ∂f/∂y
∂f/∂y
6 = 6
Calculate Magnitude |∇f|
Magnitude |∇f|
7.2111 = 7.2111
Calculate Direction
Direction = directionDeg
56.31 = 56.31

Engine last updated August 6, 2026. Checked against 2 independently-derived tests — how we verify calculators.

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