Compute the directional derivative of the following function at the indicated points in the given directions:

$\displaystyle f(x,y) = \log{\sqrt{x^2 + y^2}}, \ (x_0,y_0) = (1,0)$

$\displaystyle \bold{v} = \frac{1}{\sqrt{5}}(2\bold{i} + \bold{j})$

I did this:

$\displaystyle \nabla f(\bold{x}) = (\frac{2x}{x^2 + y^2},\frac{2y}{x^2 + y^2})$

$\displaystyle \nabla f((1,0)) = (2,0)$

$\displaystyle \nabla f((1,0)) \cdot \bold{v} = (2,0) \cdot (\frac{2}{\sqrt{5}},\frac{1}{\sqrt{5}})$

$\displaystyle = \frac{4}{\sqrt{5}}$

But I'm not sure if this is right.