# Math Help - Directional Derivatives

1. ## Directional Derivatives

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

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

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

I did this:

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

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

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

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

But I'm not sure if this is right.

2. But you are correct.
Way to go.

3. $\log{\sqrt{x^2 + y^2}} = {\textstyle\frac12}\log(x^2+y^2)$, and when you differentiate it the 1/2 cancels with the 2 from the 2x (or 2y). So your answer is twice as big as it should be. Other than that, it looks correct.

4. I see, thanks.