Derivative Intuition I'm doing self-study in Calculus, and I'm having trouble with intuitively understanding derivatives.
The definition of a derivative is: $\lim _{h\to 0}\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)$.
Also, I will note that H Does not Equal Zero.
Here is the issue I'm having: As $\lim _{h\to 0}\:$ does not equal zero, and therefore $h$ does not equal $0$, there will always be some infinitesimally small distance between $x$ and $x+h$. Therefore, the derivative is NOT a rate of change at one point, but a rate of change between $x$ and some small infinitesimal distance. I suppose one could say the slope between $x$ and $x+\text{infinitesimal}$ is close to a rate of change at a single point, but it's not the same thing. Maybe an approximation, but an approximation is not the same as the actual.
What am I missing here?
Thanks in advance for any assistance!
 A: The derivative of a function $f$ at a point $x$, or the "instantaneous rate of change of the function $f$ at $x$" is usually not a rate of change at all, in the sense that it is a quotient $\frac{f(x+h)-f(x)}{h}$ for all small values of $h\ne 0$, except in cases where the graph of $f$ is actually a line of some real slope near $x$. Instead, the derivative of a function is a limit of such quotients, taken over $h$ that approach $0$. A limit is, by definition, a number that the terms we are taking the limit of get arbitrarily close to as we send our limiting parameter to its target. Such a nice number may or may not exist!
When the limit exists, we can write $f(x+h) = f(x) + f'(x)h + \text{error}$, where $\text{error}/h \to 0$ as $h\to 0$. If we agreed to ignore the error term and then replace $f(x+h)$, whatever it actually may be, by the linear function $L(h) = f(x) + f'(x)h$, we get a function whose infinitesimal rate of change at $x$ actually is a rate of change because the graph of $L$ is actually a line of some real slope.
This is often the point-of-view that we take of the derivative of a function. The derivative of a function $f$ is the key to the best (affine) linear approximation $L$ to the function itself near the point we are taking the derivative. The instantaneous rate of change of $L$ near $x$ is a true-blue rate of change.
