Partial Derivative and Gradient Calculator
Compute partial derivatives and the gradient vector for a two-variable quadratic function at a selected point.
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Function form: f(x,y)=Ax² + By² + Cxy + Dx + Ey + F
What this calculator teaches
A partial derivative measures how a multivariable function changes when one variable changes and the other variables are held constant.
The gradient collects all first partial derivatives into a vector. It points in the direction of fastest increase.
Formula & Symbols
| Concept | Formula | Meaning |
|---|---|---|
| Partial with respect to x | f_x = 2Ax + Cy + D | Treat y as constant. |
| Partial with respect to y | f_y = 2By + Cx + E | Treat x as constant. |
| Gradient | ∇f = <f_x, f_y> | Vector of first partial derivatives. |
Step-by-Step Examples
- Let f = 2x² + 3y² + 4xy + x - 2y.
- fₓ = 4x + 4y + 1.
- fᵧ = 6y + 4x - 2.
- At (1,2), fₓ = 13 and fᵧ = 14.
Where students use this
Common Mistakes to Avoid
For fₓ, y is treated as a constant. For fᵧ, x is treated as a constant.
The derivative of Cxy with respect to x is Cy, and with respect to y is Cx.
Gradient is a vector, not a single number, when the function has more than one input.
Frequently Asked Questions
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Formula Explorer connections
Interpretation: This formula measures limiting behavior, instantaneous change, accumulation or multivariable variation. Assumption: The function must satisfy the continuity or differentiability conditions required by the method. Check domains, bounds, orientation and singularities.