Differentiability vs Having a Derivative In some calculus texts, one comes across the following statement - "A function is differentiable, if it has a derivative and a function has a derivative, if it is differentiable.". Two such texts are  - A treatise on Advanced Calculus by Philip Franklin(this book can be found on archive.org - check section 69 - Page 109) and Advanced Calculus - An Introduction to Linear Analysis by Leonard F. Richardson - Chapter 4 - Page 101.
Analytical proofs are also given to justify these statements.
This raises the question - What is the physical difference between being differentiable and having a derivative?
The book by Leonard Richardson gives some analytical explanation about the existence of a linear function, which approximates the function at points very close to x. If such a function exists, then the function is said to be differentiable(and then he goes on to prove analytically that the function has a derivative and vice versa). The problem is that the whole proof is anlaytic, and is not illustrated using a practical example. When we say that a function is differentiable, we generally check for the existence of its derivative (i.e the limit of its differential coefficient). But this text 'implies' that its possible to check the differentiability of a function by checking for the existence of such a linear function(whose slope as you've guessed becomes the derivative in the limit). So what essentially is the difference, between this approach and the standard approach? It seems purely algebraic to me....Or is there more to it?
A related question is - what's the difference between Caratheodory's definition of the derivative and the standard definition of the derivative?. The difference seems purely algebraic to me.I would like to know if there's more to it.(I think the answer to this question is closely related to the first, but since Caratheodory's definition is rarely covered in most standard calculus texts, I've decided to gain as much information as possible, on this subject.)
I am not a mathematics major, so there are limits to my theoretical thinking! I would greatly appreciate if practical examples are provided to illustrate these concepts.
 A: You're making a big deal out of nothing.  There is no a difference. A function $f: \mathbb{R} \to \mathbb{R}$ is said to be differentiable at $a$ if the following limit exists
$$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$
If the above limit exists, then it's called the derivative of $f$ at $a$, denoted as $f'(a)$.
That's all there is to it.
A: For the second question, there is no difference between the two definitions - this is a simple exercise. For the first, there is also no difference. You've defined differentiability through the exist of a linear mapping which approximates the function near $x.$ The derivative is simply the matrix which represents this function; the proof given in Franklin's book that the two are equivalent is because he essentially defines differentiable by multiplying both sides of the limit definition of a derivative by $\Delta x,$ which goes to $0,$ so it may not be entirely clear you can do this (if you're not used to this sort of thing). Anyway, there's no difference. 
For your comment above - if you define a fruit with certain characteristics as an apple, then you have a fruit which has characteristics that are easily seen to be equivalent, you need to actually prove that the characteristics are equivalent before you can conclude it's an apple. This is what they're doing - it's a trivial statement, but for the sake of a "rigorous" textbook, they've done it.
As an edit: If you have a LINEAR function that approximates the function $f$ you are looking at arbitrarily close to a point, this is the tangent line in $2$-dimensions - now, the function has a slope; this is the derivative. In higher dimensions, you will have a hyperplane that approximates this point - hyperplanes are translates of a linear mapping; linear mappings admit a matrix representation. This is your derivative.
