What is a gradient vector?

The gradient of a scalar function f(x, y) is the vector of its partial derivatives: ∇f = (∂f/∂x, ∂f/∂y). It points in the direction of steepest increase of f at a given point.

What notation does this calculator accept?

Use standard JavaScript math notation: ^ or ** for powers, * for multiplication, Math functions like sin(x), cos(x), exp(x), log(x), sqrt(x). Constants like PI are supported.

How are the partial derivatives computed?

This tool uses central finite differences for numeric approximation: ∂f/∂x ≈ [f(x+h, y) − f(x−h, y)] / (2h) with h = 1e-6. This gives high accuracy for smooth functions.

What does the magnitude of the gradient mean?

The magnitude ‖∇f‖ = √((∂f/∂x)² + (∂f/∂y)²) represents the rate of maximum increase at that point. A larger magnitude means the function changes more steeply there.

Can I use this for gradient descent?

Yes. The negative gradient (−∇f) points in the direction of steepest descent, which is exactly what gradient descent algorithms follow to minimize a function.

Gradient Vector Calculator

Name: Gradient Vector Calculator
Availability: InStock
Author: Nham Vu

Enter a function f(x,y) and a point (a, b) to instantly compute the gradient vector and visualize the direction of steepest ascent.

Function & Point

f(x, y) =

Use ^, *, sin(), cos(), exp(), log(), sqrt(), PI

x =

y =

Quick Examples

Enter a function and point, then click Compute Gradient.

Summary

Enter a function f(x,y) and a point (a, b) to instantly compute the gradient vector and visualize the direction of steepest ascent.

How it works

Enter your function f(x, y) using standard math notation (e.g. x^2 + y^2).
Specify the point (x, y) where you want to evaluate the gradient.
The calculator computes partial derivatives ∂f/∂x and ∂f/∂y symbolically via finite differences.
The gradient vector ∇f(a, b) = (∂f/∂x, ∂f/∂y) is displayed at the given point.
A canvas diagram shows the gradient arrow overlaid on a contour-like background.
The magnitude ‖∇f‖ and the unit direction vector are also shown.

Use cases

Check partial derivative calculations for multivariable calculus homework.
Find the steepest ascent direction for optimization problems.
Visualize gradient fields for machine learning intuition (gradient descent).
Verify analytical gradients against numeric approximations.
Understand saddle points and local extrema in two-variable functions.
Explore how changing x or y affects the gradient direction.

Frequently Asked Questions

Last updated: 2026-07-23 · Reviewed by Nham Vu