What is the meaning of directional derivative of a function

The directional derivative of a function $f(x,y,z)$ at a point $P$ in the direction of a unit vector $\mathbf{u}$ represents the instantaneous rate of change of the function $f$ at point $P$ as one moves in the direction of $\mathbf{u}$. It generalizes the concept of partial derivatives, which only measure the rate of change along the coordinate axes.

It is calculated as the dot product of the gradient of $f$ at $P$ and the unit vector $\mathbf{u}$:

$$D_{\mathbf{u}}f(P) = \nabla f(P) \cdot \mathbf{u}$$