What is a gradient?
The vector of first partial derivatives of a scalar function.
Gradient Calculator
Enter coefficients of a 2D quadratic form and a point to get ∇f = (fₓ, fᵧ).
Enter f and the point
Provide a, b, c for f = ax² + by² + cxy, then coordinates (x, y).
About this calculator
How to use Gradient Calculator, plus the formulas and logic behind the results.
Tool description
The gradient points in the direction of steepest ascent and its magnitude is the maximum directional derivative.
How to use
- Enter coefficients a, b, c of the quadratic.
- Enter the evaluation point (x, y).
- Select Calculate to view ∇f and its magnitude.
- Use the components as fₓ and fᵧ for related tools.
Formulas & reference
fₓ = 2ax + cyfᵧ = 2by + cx∇f = (fₓ, fᵧ)
Gradient tips
- ∇f is orthogonal to level curves of f
- Magnitude ‖∇f‖ is the steepest-ascent rate
- At a critical point, ∇f = 0
- Linear terms are omitted in this simplified f
Calculation logic
Differentiates the quadratic symbolically, then evaluates both partials at the given point.
2D educational case only—no higher-dimensional fields.
Gradient Calculator FAQ
Common questions about this calculator and how to use the results.
What f is supported?
f(x,y) = ax² + by² + cxy.
What does the magnitude mean?
The maximum rate of increase of f at that point.
Can c be zero?
Yes—then there is no mixed xy term.
Is this multivariable calculus?
Yes—introductory Calc III / vector calculus.
Are answers exact?
Numeric components are formatted for display.