Directional Derivative Calculator

A directional derivative measures the rate of change of a multivariable function in a specific direction at a given point. Unlike a partial derivative — which only measures change along a coordinate axis — the directional derivative generalizes this to any direction in the domain. It is computed as the dot product of the gradient of the function and the unit direction vector.

The gradient of a function f(x, y) is the vector of its partial derivatives: ∇f = (∂f/∂x, ∂f/∂y). The directional derivative in direction u is then D_u f = ∇f · û, where û is the unit vector in the direction of u. So the gradient encodes all directional derivative information at once.

The direction vector must be converted to a unit vector (magnitude = 1) before computing the directional derivative so that the result represents a true rate of change per unit distance. Without normalization, the result would scale with the length of the vector and would not be geometrically meaningful.

The directional derivative is largest when the direction vector aligns with the gradient ∇f. In that case, D_u f = |∇f|, the magnitude of the gradient. Conversely, it is most negative when moving directly opposite the gradient, and zero when moving perpendicular to it.

Yes. A negative directional derivative means the function is decreasing in that direction. For example, if you move opposite to the gradient, the function value falls at the rate equal to the gradient magnitude. This is the direction of steepest descent, used in optimization algorithms like gradient descent.

A partial derivative measures the rate of change of a function along one of the coordinate axes (holding all other variables constant). A directional derivative generalizes this to any arbitrary direction. Partial derivatives are actually special cases of the directional derivative — specifically when the direction vector is one of the standard basis vectors (e.g., (1,0) or (0,1)).

The calculator follows three steps: (1) it computes the partial derivatives ∂f/∂x and ∂f/∂y symbolically and evaluates them at the given point (x₀, y₀); (2) it normalizes the direction vector u = (u₁, u₂) by dividing by its magnitude; (3) it takes the dot product of the gradient vector and the unit vector to produce D_u f.

In one dimension, the directional derivative reduces to the ordinary derivative f'(x), since there are only two directions (positive or negative x-axis). For multivariable functions, the directional derivative generalizes the concept of instantaneous rate of change to infinitely many possible directions.