I don’t know how well you already know stuff so I’m going to just justify everything I can. There’s a paragraph about the gradient at the bottom. Also the logic extends so I’ll just talk about functions with 2 inputs (surfaces) instead of three. It might be more helpful. You also might need some visuals and I think there are several things online or I believe khan academy has some helpful visuals even though I’m not always a fan of their regular videos.
Consider the multivariable chain rule. Recall that it works intuitively because the change in z arises from the combined changes in x and in y. Also you can try some practical thought experiments help make it clear, all with the basic idea that for $z(x(t),y(t))$, the change of the height z as you go through time (or more generally some other parameter) is the amount it changes due to x plus the amount it’s changing due to y since they are independent. You could readily substitute amount and rate in that previous sentence, which yields the multivariable chain rule, as was desired.
Now, this $f(x(t),y(t))$ can be interpreted as a path above a graph on the xy plane, which goes along the surface $f$. When the graph (x,y) is a straight line and you move along it with constant speed, you have the derivative as you move in that direction (not just x or y, which are the partial derivatives). If you move at different speeds, like twice the speed, the height changes twice as much in the same amount of time so the directional derivative also doubles. Directional derivatives, however, we take when moving at a constant speed (so they’re like a regular partial derivative but moving in a different direction), thus the vector is normalized. That’s the interpretation of not normalizing the vector (which is just moving at a different speed) cause I don’t feel like normalizing this: suppose $\frac{dy}{dt}=2$, and $\frac{dx}{dt}=3$ (I’ll try to look up and make the partial derivative symbol in there later). As you move along the vector 2,3 in a unit of time, the rate of change of z is these substituted into the multivariable chain rule, which is algebraically equivelant to the gradient dot the vector you move along in a period.
Additionally, the dot product is saying how much length of one vector there is per length of the other along it’s direction, because this is equivalent to projecting one vector onto another and multiplying their lengths. Note: it doesn’t matter which is projected onto which because the dot product is symmetric. Anyways, locally the gradient is the direction of steepest ascent, and dotting a vector with it is just saying how much is this unit motion lined up along this direction of main (positive) ascent, the quantity you’re measuring. This comparison to the gradient is logically the same as saying how much of the increase (gradient) is in a particular direction, hence, directional derivative.
*The gradient is how a function ascends, ie a certain amount in one direction plus a certain amount in the other(s), or: $i\frac{df}{dx}+j\frac{df}{dy}$ (again, sorry about the latex, and that’s meant to be i hat and j hat). Viewed in this light, it only makes sense that “how it’s ascending” ascends in the direction of steepest ascent. It’s just as a gradient of a single variable function is a vector telling you loosely “how it’s changing”, and if you can connect this to the idea that the vector (which is just a scalar in this case) points in the direction of increase according to how it’s increasing you can see the analogy for higher dimensional gradients. Of course you could also prove this analytically in several ways, one of which is working backwards and optimizing the angle of the directional derivative (which weirdly isn’t circular reasoning and doesn’t require you to already know about the gradient).
I haven’t thought through this too much but since every analytic function is has a tangent plane you could figure it out by doing it on a plane.
And again, choosing some real world or more abstract variables with this and doing a thought experiment is fun. And like Mohammad said, like normal slope, it’s a scalar.
Your proof is correct but is, in my opinion, missing a few words here and there explaining why your procedure is correct.
As an example : it seems to me that you're implicitly using the property that the directional derivative $\partial_v f$ at a point $a$ is the dot product $\nabla f (a) \cdot v$. This is not always the case ! But it is true when the function $f$ is differentiable at $a.$ So your proof should at least contain the words " Since $f$ is differentiable at ..."
Similarly you could be elaborate a bit more on certain parts:
The minimum directional derivative is:
$$−|\nabla f(2,1)|=−|⟨1,−1⟩|=−\sqrt 2$$
Why is this true ? You don't need the write a huge paragraph of course ;) but a few extra words here and there would make your proof far more readable.
Best Answer
Usually $\nabla$ only acts on the object immediately to its right. Another indication that that can't be the correct interpretation is that "$\cdot$" is an operation on two vectors, and $f$ is a scalar, while $\nabla f$ is a vector, so $f \cdot u$ doesn't make sense.
Yes, the convention with $\nabla$ can be a bit confusing, so you may prefer to write the directional derivative as $\mathbf{u} \cdot \nabla f$, or $(\nabla f) \cdot \mathbf{u}$. (You also find it written as $(\mathbf{u} \cdot \nabla)f$ to emphasise that $\mathbf{u} \cdot \nabla$ is the directional derivative operator, which sends scalar fields to scalar fields.) If you think an expression can be ambiguous, it's always best to bracket it carefully, just as $\sin{x}y$ could mean either $(\sin{x})y$ or $\sin{(xy)}$.
(The object you have written on the right-hand side of the last displayed equation is actually the divergence of the vector field $f\mathbf{u}$, i.e. $\nabla \cdot (f\mathbf{u})$.)