directional derivative

The rate of change of $f$ in an arbitrary direction, specified by a unit vector, generalizing the partial derivatives.

The directional derivative $D_{\mathbf{u}}f = \lim_{t\to0}[f(x_0+ta,y_0+tb)-f(x_0,y_0)]/t$ generalizes the partial derivative to an arbitrary direction $\mathbf{u}=\langle a,b\rangle$ , with the two partial derivatives falling out as the special cases $\mathbf{u}=\langle1,0\rangle$ and $\mathbf{u}=\langle 0,1\rangle$ . The unit-length requirement on $\mathbf{u}$ isn't cosmetic: only when $\langle a,b\rangle$ has length 1 does $t$ measure actual distance traveled, making the rate comparable across different directions.

Examples

For $f(x,y)=x^2y$ , $D_{\mathbf u}f(1,2)=16/5$ in the direction from $(1,2)$ toward $(4,6)$ .