What does the derivative of a function at a point describe? I understand that the derivative of a function $f$ at a point $x=x_{0}$ is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where $\Delta x$ is a small change in the argument $x$ as we "move" from $x=x_{0}$ to a neighbouring point $x=x_{0}+\Delta x$. 
What confuses me is how to interpret its meaning correctly, that is, what does the derivative $f'(x_{0})$ actually describe? 
On Wikipedia it says that "the derivative of a function quantifies the rate at which the value of the function changes as we change the input" (or words to that effect). However, the function has a particular constant value, $f(x_{0})$ at a given point $x=x_{0}$ so how can one meaningfully discuss the rate at which the value of the function is changing at that point?
Would it be correct to interpret the derivative of a function at a point as describing how "quickly" it's value changes as we move from that point to (infinitesimally close) neighbouring points? (As such in the example above, in moving from the point $x_{0}$ to $x_{0}+\Delta x$ the value of the function $f$ changes by an amount $f'(x_{0})\Delta x$ for infinitesimally small change $\Delta x$). Is it then simply that the value of the derivative at that point equals the slope of the tangent line to the the function (curve) at that point? (In general then, the derivative of a function is itself a function whose value at each point equals the slope of the tangent line to the curve at that point).
 A: I think your hung up on the idea that a rate needs an associated time interval that it applies to. This isn't true, I think most people probably intuitively think this but consider the following scenario. 
Think about a ball being dropped from a building. At every possible moment the ball is traveling a different velocity because it is constantly accelerating due to the force of gravity. Velocity, of course is a rate, specifically, its the rate at which the position of the ball is changing. So, no matter how hard you look, there is no interval, no matter how small, at which the velocity of the ball is a specific number. It will only be traveling at a specific velocity at one moment in time. So, the derivative of the balls position, at some time t is that one exact point in space where the ball will be traveling at that velocity. See? no interval needed.
If it helps you to think about the rate as a difference over some infinitesimal time than go for it, the definition of a limit above says that a specific moment and this infinitesimal difference are the same thing. Thats part of why the results of calc are so cool. I think once you start to really feel like those two are the same thing you wont need to associate rates with an interval to which they apply. Best of luck.
A: Your understanding of the derivative is correct. The only thing I would slightly reword is instead of saying
"in moving from the point $x_{0}$ to $x_{0}+\Delta x$ the value of the function $f$ changes by an amount $f'(x_{0})\Delta x$ for infinitesimally small change $\Delta x$"
I would write
"in moving from the point $x_{0}$ to $x_{0}+\Delta x$, the amount that the value of the function $f$ changes approaches $f'(x_{0})\Delta x$ as $\Delta x$ approaches 0"
The reason I say this is because in many cases, there is no actual value of $\Delta x$ such that the change in $f$ from $x_{0}$ to $x_{0}+\Delta x$ actually equals $f'(x_{0})\Delta x$, as there is no actual number that is infintesimally small.
A: Your understanding of the interpretation of the derivative as a rate of change is correct but there is a subtlety.
Let's look at $f(x)=x^2$ where $f:\mathbb R \to\mathbb R$.
$f'(x)=2x$ so $f'(1)=2$.
Does this mean that the derivative $f'(1)$ is the real number $2$?
It does not. The $2$ in this case is the function that takes $x$ to $2x$. Only when we speak of the derivative we usually say it takes $dx$ to $2dx$.
In fact, $f'(x_0)$ is itself a linear function whose graph is the tangent line to the graph of $f$ at the point $x_0$. The input to this function is called the tangent space and $x_0$ is the origin of this input space.
