Difference between "Differentiability" and "Differentiation" . This is a question of which i have not got a clear visualization still , so thought of asking for your views on it.
In the textbook, there are two separate chapters... one is "Differentiability" and the other is "Differentiation"... My question is simple : 

What is the difference between the two?

Why i asked this is because when we see the proofs of differentiation of any basic functions (be it $\cos x$, $\sin x$, $e^x$, $\tan x$, etc.) ,they are proved by the method "differentiation from first principles" using the basic definition of Differentiability at a point. 
I tried to read the answers on this forum for similar questions like this previously asked, but the answers to them were  of higher understanding level which i have not been enlightened yet. Is there any simple explanation available for this..If anyone can help me understand the difference between the two in simple words, kindly do so... and thanks a lot for all your help.
 A: Ok, so the intuitive pictorial idea of a derivative of a function $f$ at a point $x$ is taking the slope of the graph of $f$ at the point $x$. Now this is fine for guiding intuition but it isn't mathematically rigorous so let's try to be more rigorous.
So firstly what we do is we try to approximate the slope at a point, you may have heard of secant lines? That is what we are doing here. So to approximate the slope we take a small value $h$ and we take the slope between $f(x)$ and $f(x+h)$. Remember that gradient is "rise over run?" so our approximation is our rise $f(x+h)-f(x)$ over our run $(x+h)-x=h$. Hence our approximation is:
$$
\frac{f(x+h)-f(x)}{h}
$$
I hope you recognize this equation.
Ok so we have an approximation, now the idea is that if our function $f$ is smooth enough at $x$ then as $h$ gets smaller, this approximation is going to get better. If this is the case wouldn't we want to take $h=0$ to get the best approximation? Unfortunately we have a problem here, if you substitute $h=0$ into the approximation above we get $\frac{0}{0}$ which is bad. So what are we going to do?
Well what we do is hope that as $h$ gets close to 0, that our approximation starts to settle down and get close to the gradient we are trying to approximate. To this end we ask does
$$
\lim_{h\rightarrow 0}
\frac{f(x+h)-f(x)}{h}
$$
Exist? If it does, then our approximation is good, and we say that $f$ is differentiable at $x$. And we call the value $
f'(x)=\lim_{h\rightarrow 0}
\frac{f(x+h)-f(x)}{h}
$ the derivative of $f$ at $x$.
If the limit does not exist then it means that our function is not smooth enough to take good approximations of the gradient, and if we can't approximate it well we say it doesn't exist, and we say $f$ is not differentiable at $x$.
So a function is differentiable if the limit above exists, and the derivative is the value of the limit.
This clear things up?
A: A function is differentiable if we can differentiate it.
Differentiability refers to the existence of a derivative while 
differentiation is the process of taking the derivative. 
It is like the difference between perceptibility and perception. 
An object is perceptible if we can perceive it. 
Perceptibility refers to the ability to perceive an object while perception is the act of perceiving the object. 
