Multivariable calculus, using unit vector?

Question

Suppose we want the derivative of $f(x,y)$ in the direction of $\langle 1,1\rangle$. We must convert this to a unit vector: $\langle 1/\sqrt2, 1/\sqrt2\rangle$ in order to use the formula. So I don’t find it intuitive that the partial $x$ must be multiplied by $1/\sqrt2$ and the partial $y$ must be multiplied by $1/\sqrt2$, take the sum, and that’s your derivative in the direction $\langle 1, 1\rangle$.

Answer 1

The directional derivative is the rate of change of your function at a given point when you move along the given direction. You want to find the linear change when you move in the given direction one unit of length. That is why you normalize your direction vector to make its length $1$.