TITLE: Differentiation: exact or an approximation?
QUESTION [7 upvotes]: I understand that a derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. This makes sense, since we would take the limit of the interval over which the derivative is computed as the interval tends to zero.   
In other words, take the limit of the average rate of change as the interval over which the average is computed tends to zero. 
But I read today that "differentiation is a method to find an exact value for the rate of change at any given value of $x$". 
So is it a linear approximation or is it an exact value? 
Thanks.

REPLY [2 votes]: Another point of view on differential calculus is through its relation to integral calculus -- specifically, the fundamental theorem of calculus, which says
$$ f(b) - f(a) = \int_a^b f'(x) \, dx $$
Differential calculus can be seen as the business of attaching values to a point that quantifies behavior near that point. The simplest example of this is simply the value of a continuous function: if $x$ is near $10$, then $x^2$ is near $100$, for suitable meanings of the words "near" and "near". (the precise statement is the $\epsilon-\delta$ definition of limits)
To capture more information, we introduce the differential of a function. In this case, the differential of $x^2$ is given by $d(x^2) = 2x \, dx$. The value of this differential at $x = 10$ is $20 \, dx$.
For single-variable differentiable functions, $df(x)$ is always proportional to $dx$, which means we can define a function $f'(x)$ to be the constant of proportion: $d(f(x)) = f'(x) \, dx$. This allows us to talk about the derivative $f'(x)$ as an ordinary function, allowing us to defer trying to give a precise meaning to the notion of a differential.
The integral is the tool to extract "ordinary" information out of the "differential" information; the fundamental theorem of calculus above shows how the actual change in $f$ is recovered by the integral of $d(f(x))$, and therefore $d(f(x))$ is a sort of differential version of the notion of the "change of the value of $f(x)$".