Let  be a given function of . Given 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  be 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 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.

Now we write in a table the notations for derivative of  used by different mathematicians.

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 means 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.
This gives the derivative required employing the above mentioned four steps in determining the derivative means finding the differential coefficient “by definition” or “by first principle” or “by ab-initio method”.