Can someone help me compute the following directional derivative using the formal definition: $ D_u = \lim_{t\to 0 } \frac{f(x_0 +tu)-f(x_0) }{t}$ ?
the function is: $ f(x,y)= \arctan(x^2 + y^2 ) $ , at the point $ x_0 = (1,1)$, in the direction $u = (1,-2) $ .
The answer should be $ -\frac{2}{5 \sqrt5 } $ .
After putting it all together I received the limit of the following expression: $\frac{\arctan ((1+\frac{t}{\sqrt{5} })^2 +(1-\frac{2t}{\sqrt{5}} ) ^2 ) - \arctan(2)}{t}$ and I have no idea how to compute it.
Thanks in advance !