Directional derivatives are one way to find a tangent vector to a surface.
A tangent vector to a surface has a slope (rise in $z$ over run in $xy$) equal to the directional derivative of the surface height $z(x,y).$
If $\mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k}$ is tangent to the surface $z(x,y)$ at $(x,y)$, then $$ \frac{c}{ | a \mathbf{i} + b \mathbf{j} | } = \frac{dz}{ds} \Bigg|_{a \mathbf{i} + b\mathbf{j}}$$
To find a tangent vector, choose $a,b,c$ so that this equality holds.
