Second order directional derivative?

If $\displaystyle \frac{\partial f}{\partial v} (x,y)= (f_x,f_y) \cdot \frac{v}{|v|}$

is $\displaystyle \frac{\partial f^2}{\partial v^2} (x,y)= (f_{xx},f_{yy}) \cdot \frac{v}{|v|}$ ?