directional derivative

A partial derivative asks how fast the function changes if you march along a coordinate axis. A directional derivative removes that restriction: pick any direction you like — northeast, or any compass heading — and ask how fast the function changes as you step that way. It is the slope of the function along an arbitrary straight line through the point.

Formally, the directional derivative of f at a in the direction of a unit vector u is the limit of (f(a + t u) - f(a))/t as t -> 0, often written D_u f(a). The partial derivatives are the special cases where u is a coordinate vector e_i. The definition reduces a multivariable rate of change to an ordinary one-variable derivative along the line t -> a + t u.

If f is differentiable at a, then all directional derivatives exist and obey the clean formula D_u f(a) = grad f(a) . u — they are linear in the direction. But beware: a function can have a directional derivative in every direction at a point and still fail to be differentiable or even continuous there. Existence of directional derivatives is weaker than differentiability.