Second order  directional derivative

I understand how to calculate second order directional derivative. I want to get better understanding of the formula of it.
So first order directional derivative of f(x,y) in direction of 'u' is:
$$D_uf(x,y) = \vec{\nabla}f\cdot{u}=f_x(x,y)a + f_y(x,y)b$$
So if I want to calculate second order derivative (in direction of 'u') I will have:
$$D_u(D_uf(x,y)) = \vec{\nabla}(f_x(x,y)a + f_y(x,y)b)\cdot{u} =$$
$$(f_{xx}(x,y)a + f_{xy}(x,y)b)a + (f_{yx}(x,y)a + f_{yy}(x,y)b)b$$
Or what?) I think I'm wrong here, please explain.


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Kav10 Kav10
  • is $\frac{\partial}{\partial y} \left( f_x(x,y) \cdot a + f_y(x,y) \cdot b \right)$ the final state, or I can somehow calculate it?

    • Kav10 Kav10

      Pretty much, yes. You can simplify it a bit more.

    • Kav10 Kav10

      a \cdot \left( a \cdot \frac{\partial f_x}{\partial x} + b \cdot \frac{\partial f_y}{\partial x} \right) + b \cdot \left( a \cdot \frac{\partial f_x}{\partial y} + b \cdot \frac{\partial f_y}{\partial y} \right)

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