Answer
$$\dfrac{-11}{25\sqrt{34}}$$
Work Step by Step
Our aim is to determine the directional derivative. In order to find it we have to use the expression:
$D_uf(x,y)=f_x(x,y)m+f_y(x,y)n$
or, $D_uf(x,y)=\nabla f(x,y) \cdot u=\nabla f(x,y) \cdot \dfrac{v}{|v|}$
Given: $f(x,y)=\dfrac{x}{x^2+y^2}$
From the given data, we have : $(x,y)=$$(1,2)$
$D_uf (1,2)=(\dfrac{3}{25})(\dfrac{3}{\sqrt{34}})+(\dfrac{-4}{25})(\dfrac{5}{\sqrt{34}})=\dfrac{-11}{25\sqrt{34}}$
