Gradient Calculator

The gradient of a scalar field f is a vector pointing in the direction of the steepest increase. At any point, it tells you which direction to step to increase f most rapidly. The magnitude of the gradient equals the slope in that steepest direction.

For f(x,y), the gradient is the vector: del f = (df/dx, df/dy)

Each component is a partial derivative — the rate of change along one axis with the other held fixed.

This calculator handles three function forms:

Separable sum: f(x,y) = ax^p + by^q. Partial derivatives are df/dx = pax^(p-1) and df/dy = qby^(q-1). These are independent of each other because x and y appear in separate terms.

Product: f(x,y) = ax^p * y^q. Now both variables mix: df/dx = pax^(p-1)y^q and df/dy = qax^py^(q-1). The y term survives in the x-derivative and vice versa.

3D separable: f(x,y,z) = ax^p + by^q + cz^r. The gradient becomes a 3D vector with a third component df/dz = rcz^(r-1).

Worked example: f(x,y) = 3x^2 + 2y^3 at (1, 2). df/dx = 6x = 6(1) = 6. df/dy = 6y^2 = 6(4) = 24. Gradient = (6, 24), magnitude = sqrt(36 + 576) = sqrt(612) = 24.74. From point (1,2), the function increases fastest in the direction (6, 24) — mostly along the y-axis.

The gradient is always perpendicular to the level curves of the function. This is why gradient descent in machine learning follows the negative gradient: it is guaranteed to move downhill at every step.