Partial Derivative Calculator

The Partial Derivative Calculator computes the partial derivatives of a two-variable quadratic function $$f(x,y) = ax^2 + bxy + cy^2 + dx + ey + f$$ at any specified point. It calculates both first-order partial derivatives ($$\partial f/\partial x$$ and $$\partial f/\partial y$$), all three second-order partial derivatives, the gradient magnitude, and the Hessian determinant with classification. This is a fundamental tool for multivariable calculus, optimization, and surface analysis.

Partial derivatives extend the concept of a derivative to functions of multiple variables. Unlike ordinary differentiation, where you differentiate with respect to the only independent variable, partial differentiation holds all variables constant except the one you are differentiating with respect to. The notation $$\frac{\partial f}{\partial x}$$ means "differentiate f treating y as a constant," and $$\frac{\partial f}{\partial y}$$ means "differentiate f treating x as a constant."

The theory of partial derivatives was developed in the 18th century by Leonhard Euler, Alexis Clairaut, and Joseph-Louis Lagrange. Clairaut's theorem, a key result, states that for functions with continuous second partial derivatives, the mixed partials are equal: $$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$$. This symmetry is confirmed by the calculator — both mixed partials equal $$b$$.

For the quadratic function $$f(x,y) = ax^2 + bxy + cy^2 + dx + ey + f$$, the partial derivatives are exact and elegant:

$$\frac{\partial f}{\partial x} = 2ax + by + d, \qquad \frac{\partial f}{\partial y} = bx + 2cy + e$$

The gradient vector $$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$$ points in the direction of steepest ascent of the surface. Its magnitude tells you how rapidly the function is changing. The gradient is zero at critical points, where the surface has a local maximum, local minimum, or saddle point.

The Hessian matrix of second partial derivatives determines the nature of critical points. For our quadratic, the Hessian is constant: $$H = \begin{pmatrix} 2a & b \\ b & 2c \end{pmatrix}$$. Its determinant $$D = 4ac - b^2$$ classifies the surface: $$D > 0$$ with $$a > 0$$ gives a local minimum (elliptic paraboloid opening up), $$D > 0$$ with $$a < 0$$ gives a local maximum (opening down), $$D < 0$$ gives a saddle point (hyperbolic paraboloid), and $$D = 0$$ is degenerate.

Applications of partial derivatives span every quantitative field. In thermodynamics, partial derivatives relate pressure, volume, and temperature changes. In machine learning, gradient descent uses partial derivatives to optimize model parameters. In economics, partial derivatives compute marginal productivity of labor and capital. In fluid dynamics, partial derivatives appear in the Navier-Stokes equations. Enter your function coefficients and evaluation point below.