Partial Derivative Calculator

This calculator evaluates the partial derivatives of f(x,y) = axy + bx² + cy² + d — the coefficients labeled a, b, c, and d — at a chosen point (x, y), reporting ∂f/∂x, ∂f/∂y, f(x,y) itself, and the gradient magnitude. The two partials are cleanly separated by code, not coincidence: ∂f/∂x is computed as coeffXY * yValue + 2 * coeffX * xValue, a line that never reads Coefficient of y² (c) at all, so c is genuinely inert on ∂f/∂x for any value it holds; symmetrically, ∂f/∂y never reads Coefficient of x² (b).

Coefficient of x² (b) and x Value show an identical measured effect on ∂f/∂x under a small nudge, because only their product, 2 * coeffX * xValue, appears in the formula — scaling either factor by the same percentage moves that product by the same amount, so this calculator does not single out one of the two as dominant. What no sensitivity check here can confirm, even though it's a basic calculus fact: Constant (d) never appears in either partial-derivative formula at all (its whole contribution to f(x,y) has zero slope), but it also defaults to 0, and a nudge probe that scales a value by a fixed percentage of itself can't move an input that starts at zero — so this genuinely true relationship falls outside what the mechanical check can verify.