Derivative of a Function
Let be a given function of . Give to a small increment and let the corresponding increment of by , so that when becomes , then becomes and we have:
Dividing both sides by , then
Taking limit of both sides as
Thus, if is the function of , then is called the derivative or the differential coefficient of the function or the derivative of with respect to and is denoted by , , or .
Note: It may be noted that the derivative of the function with respect to the variable is the function whose value at is provided the limit exists, is called the derivative where is the increment.
Below we list the notations for derivative of used by different mathematicians in a table.
Name of Mathematician

Leibniz

Newton

Lagrange

Cauchy

Notation for Derivative

or 
Remarks:
I. The student should observe the difference between and . Where is the quotient of the increment of and that is its numerator and denominator can be separated, but is a single symbol for the limiting value of the fraction when and can be separated.
II. attached to any function meaning its differential coefficient with respect to .
III. The phrase “with respect to” will often be abbreviated into w.r.t
Differentiation:
The process of finding the differential coefficient of a function or a process for finding the rate at which one variable quantity changes with respect to another is called differentiation.
Four Steps in Differentiation:
In the given function
1. Change to and to
i.e.
2. Find by subtraction
i.e.
3. Divide both sides by
i.e.
4. Find the limit of where
i.e.
Employing the above mentioned four steps in determining the derivative means finding the differential coefficient “by definition” or “by first principle” or “by abinitio method”.