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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.

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    $\begingroup$ Differentiability refers to the ability to take the derivative of a function and differentiation is the act of taking the derivative. $\endgroup$ – John Douma Jun 15 '13 at 20:19
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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?

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  • $\begingroup$ Thanks for your feedback. So,here you took the right hand derivative. And we can take the left hand derivative too on similar lines. So, if lim(x->a-) f'(x) = L.H.D. and lim(x->a+) f'(x) = R.H.D. (here i took x=a as the value of x in L.H.D. and R.H.D., with "a" being the point where we want to check the differentiability of the function) ,then we can say that the function is differentiable at the point "x=a". This is what u are saying right??? If i have not got it right,lemme know... and thanks once again for taking time to write :)) $\endgroup$ – under root Jun 16 '13 at 20:42
  • $\begingroup$ and if the above conditions are satisfied, we must have RHD. = L.H.D , and we are done.... right? $\endgroup$ – under root Jun 16 '13 at 21:03
  • $\begingroup$ So, we can then say by the abuse of language : "differentiation of a function in a domain is found out by checking if R.H.D. and L.H.D. exists for every point in the domain, and hence if thats the case, then the function is differentiable.".... in general, we get differentiation of a function using differentiability ... am i right?? sorry u have to read a lot... i wanted to make clear my understanding of the concept.. :)) $\endgroup$ – under root Jun 16 '13 at 21:07
  • $\begingroup$ let me know if i have understood it the right way whenever u get time... and if i m wrong somewhere, plz point out my mistake..thanks yet once again... $\endgroup$ – under root Jun 16 '13 at 21:08
  • $\begingroup$ Actually I didn't mention one sided limits our "small $h$" can be negative. If you want to talk about left and right side limits that works too but you need the limit to exist on both sides, and for the limits to be the same to say the function is differentiable. To see why we need this you can look at $f(x)=|x|$ at $0$. $\endgroup$ – James Jun 16 '13 at 21:32
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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.

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    $\begingroup$ @under-root: Glad to help. $\endgroup$ – user26872 Jun 16 '13 at 21:56

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