What is differentiation?

Differentiation in math terms is the mathematical procedure of taking the derivative of a function. A derivative of a function is a function that gives the slopes of the tangent lines to each point of the curve representative of the function on a graph. In a physics perspective, the derivative of a function for distance is the velocity and the second derivative of distance or the first derivative of velocity is the acceleration.

---

The main aim of Differentiation is to find how quickly a mathematical function changes at a particular place.
If you know about the behavior of the function throughout its span,
differentiation will show its rate of change.

Of course when you differentiate the function at a point it will give the slope of the tangent that you draw at that point.

---

Both of the above are correct, but just one more, derivative does not always exist. For the derivative to exist at point x = a, the limit of (f(x) - f(a)) / (x - a) as x approaches a must exist.

We say a function is differentiable at x = a if that limits exist.
We say a function is differentiable if it is differentiable at every point a in its domain.
Remark, it is only important that the limit exist, that (f(x) - f(a)) / (x - a) doesn't have to be defined at x = a. Usually, it's 0/0, which is not defined.

(It is not difficult to prove that it is equivalent to the limit of (f(a + h) - f(a)) / h as h approaches 0)