What is the use of differentiation? I tried to know from lot of books. But they always give only the formulas for differentiation. I want to know why do we differentiate the expressions?
 A: The intuition behind differentiation is to try and replace a non-linear function (ex: $\ln$) with a linear function (ex: $z \mapsto az+b$). Now there is no hope to do that in any significant and useful way globally (i.e. replace the function on all of its domain of definition) but it works locally, i.e. near an arbitrary point. 
Example : $\ln$ taken near $x=1$ "locally looks like" $x \mapsto x-1$.
Differential calculus is the study of "what you can say about your original function" when you remember only this partial, local information that the derivative is (as it turns out, you can say a lot, and this is why differential calculus is important!).
example : a function on $\mathbb{R}$ is increasing if and only if its derivative is positive everywhere. The fact that the derivative is positive at a certain point $x$ means that $f$ is "increasing near $x$".
A: In addition to the immediate use of derivatives and integrals to study curves (tangents, areas, etc), the big boost to "calculus" was Newton's formulation of basic physics, especially planetary motion, in terms of "differential equations", that is, equations positing relations between functions describing position, velocity, acceleration, and such. Subsequently, and still today, differential equations are an incredibly useful model for many physical phenomena, as well as for more abstract processes.
Even with the advent of the notion that physical space is not a continuum, but has a grainy or foamy quantum nature, modeling it by a continuum is apparently very convenient for human consumption, especially if not aided by modern computers.
So, by now, we know that derivatives play an important role in our description of things in the world, EDIT: and not only description, but prediction, importantly.
