Finding unit vector from directional derivative

Question

I was given this question where $f(x, y) = x^2 + xy^2$ and I have to find the unit vector at the point $(2, 1)$ when the directional derivative has the value of $2.$ I am still quite confused on how to do this as I am still a beginner :( pls help

Answer 1

The gradient of $f$ in $p(2,1)$ is $\begin{pmatrix}5\\4\end{pmatrix}$. As the directional derivative of $f$ in $p$ in direction of the unit vector $\begin{pmatrix}u\\v\end{pmatrix}$, that is, $u^2+v^2=1$, is the product of the gradient and the direction vector, we derive that $5u+4v=2$. Now solve for $u$ and $v$.