Explain partial derivatives v2

I will give you a part of text taken from the book. Please explain the part where author explains directional derivative:
For functions with multiple inputs, we must make use of concept of partial derivatives. The partial derivative $$\frac{\partial}{\partial{x_i}}f(x)$$ measures how f changes as only the variable $$x_i$$ increases at point x. The gradient generalizes the notion of derivative to the case where the derivative is with respect to a vector: the gradient of f is the vector containing all of the partial derivatives, denoted $$\nabla_xf(x)$$
Element i of the gradient is the partial derivative of f with respect to $$x_i$$
Until now everything is clear.
But here comes the part which I don't understand :
The directional derivative in direction u (a unit vector) is the slope of the function f in direction u. In other words, the directional derivative is the derivative of the function $$f(x+\alpha{u})$$with respect to $$\alpha$$
evaluated at $$\alpha = 0$$
Using the chain rule, we can see that $$\frac{\partial}{\partial{\alpha}}f(x+\alpha{u})$$ evaluates to $$u^T\nabla_xf(x)$$ when $$\alpha = 0$$

So please explain:
1. What does $$f(x+\alpha{u})$$ represent (I don't really understand the idea behind x + au)?

2. According to the chain rule we have to have a function where f(x,y) and x is a function of some other variable, together with y, for example x(t), y(t), so f is also a function of t and I can find df/dt. But in case of $$f(x+\alpha{u})$$ I don't really understand why does Chain rule work here, so please explain.

3. And how does derivative of 2 evaluate to $$u^T\nabla_xf(x)$$

P.S. If you spend more time on explaining than equals to 10$ I will tip you.


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Mathe Mathe
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