I have to evaluate the directional derivative of the function $$f(x,y)=\frac{xy}{x^{2}+y^{2}+1}$$ in the direction of the vector $$u(t)=(t,1+2t)$$ at the point $$P(1,-1)$$
I know $$\nabla f(1,-1)=\left(-\frac{1}{9},\frac{1}{9}\right)$$ but my question is: how to normalize the vector $u$? I found out that for $t=1$ $$u(1)=(1,3)$$ and then $$D_{u}f=\left(-\frac{1}{9},\frac{1}{9}\right)\cdot\frac{1}{\sqrt{10}}(1,3)=\frac{2}{9\sqrt{10}}$$ but I'm not sure if this is the right way to do it.
What was the original statement of the problem?
If we interpret $u(t)$ as a vector, then it's not a single vector, but rather an infinite family of different vectors — one for each real value of $t$. And we can find the corresponding family of answers, but I doubt that's what this question means.
It looks more likely that the given $u(t)$ represents a line, and the question is to find the directional derivative in the direction of the directional vector of this line. From the given vector-parametric equation of the line, its directional vector is $\mathbf{u}=(1,2)$, and then you proceed as you did.