What does mean the exact value of derivative i'm starting my calculus's journey and i have a question.
What does mean the exact value of a derivative 
Take an easy example we have a derivative of $f(x)=x^2$, that is $f'(x)=2x$.
Someone would say that for every $x$, $f'(x)$ is tanget line of $f(x)$ with respect to $x$
And this is true, graphically it looks like that, i don't have doubts here, it is just a graphical representation.
Next one would say that $f'(x)$ is how fast the function changes.
But what does mean? for a linear function it describes how the function changes, say we have a linear function $g(x)$, if we change the argument $x$ by $h$, then $g(x)$ value changes by $h*g'(x)$. if we go by analogy, then if $f(x)$ changes by $h$ then $f(x)$ changes by $h*f'(x)$, but for every $x$ it would be only a approximation.
Let's compare $f(x)$ to car's traveled distance, then $f'(x)$ would be iinstantaneous speed of the car. Before derivatives car's speed was saying me that "How far i would travel by some amount of time, if speed doesn't change", now i'm confused. 
I have another question related with the topic.
We know that chain's rule is defines as $'(f(g(x))) = f'(g(x))*g'(x)$. But i can't undestand why is that. I know, there are proofs, but they to hard for me at this moment and i can't read the intuition that stands behind chain's rule.
Thx to every response.
 A: The geometric idea is what you stated about the tangent line. The analytic idea is:
$$f(x+\Delta x)=f(x)+f'(x) \Delta x + r(x,\Delta x)$$ 
where $r(x,\Delta x)$ is small if $\Delta x$ is small. That is, $g(\Delta x) = f(x+\Delta x)$ is nearly a linear function if $\Delta x$ is small enough. "Small" and "small enough" are quantified by the definition. Specifically, if you rearrange the definition of the derivative, you get:
$$\lim_{\Delta x \to 0} r(x,\Delta x)/\Delta x = 0.$$
To think about the chain rule, let's consider a physical situation. Let's say we're moving along a path which is the graph of some function $y(x)$. Then at each time $t$ we are at the point $(x(t),y(x(t))$. The chain rule tells us how to compute the $y$ component of our velocity:
$$\frac{dy}{dt} = \frac{dx}{dt} \frac{dy}{dx}$$
This says that the $y$ velocity is the $x$ velocity times the slope of the curve that we are following at the point where we currently are. Equivalently, we can look at
$$\frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$
This says that if we follow a curve $(x,y(x))$ with a parametrization $(x(t),y(t))$, the slope of the curve is the ratio of the $y$ velocity to the $x$ velocity.
