Finding the direction in which the derivative is exactly $2$

Question

I found the gradient taking partial derivatives and got $\nabla f = (1,2)$. I know that the directional derivative has to equal $2$ so if I set $(1,2)\cdot (x,y) = 2$ , where $(x,y)$ is a unit vector, I get $x + 2y = 2$. I'm not sure how to proceed from here.

Answer 1

A "direction" is nothing but a unit vector, since the way in which a vector is oriented depends only upon the way in which its unit vector is oriented.

Therefore, you may assume that $(x,y)$ has unit length i.e. $x^2+y^2 = 1$, and then find $x,y$ , being two equations of two unknowns.

(Note : the computations performed by you are correct till the end of your question)

Answer 2

You are right all along. Now, it is a bit of time solving the following system of variables:$$\begin{cases}x+2y=2\\x^2+y^2=1\end{cases}$$which has two answers $$(x,y)=(0,1)\\(x,y)=(0.8,0.6)$$