The Directional Derivative Calculator helps measure how quickly a multivariable function changes at a point when you move in a chosen direction. Unlike a partial derivative, it is not restricted to moving along one coordinate axis.
The Directional Derivative
For a differentiable function f, the directional derivative in unit direction u is Dα΅€f = βf Β· u. The gradient βf contains the partial derivatives, and the dot product combines them according to the chosen direction.
Prepare the Direction
A direction vector must be converted to a unit vector before the standard directional derivative formula is applied. If v is the supplied direction, its unit form is u = v/|v|. This normalization prevents the length of the direction vector from changing the rate being measured.
Example
For f(x,y) = xΒ² + yΒ², the gradient is (2x, 2y). At (1,2), the gradient is (2,4). If the movement direction is (3,4), the unit vector is (3/5,4/5), so the directional derivative is 2(3/5) + 4(4/5) = 22/5.
Interpreting the Result
A positive value means the function increases in the selected direction, while a negative value means it decreases. A value near zero indicates little first-order change in that direction. The largest directional derivative occurs in the direction of the gradient.
Good Practice
Check the point, function, and direction separately before calculating. If the tool accepts a direction vector directly, confirm whether it performs normalization automatically or expects a unit vector.