Derivative of a Function

Let y = f(x) be a given function of x. Give to x a small increment \Delta x and let the corresponding increment of y by \Delta y, so that when x becomes x + \Delta x, then y becomes y + \Delta y and we have:

y + \Delta y = f(x + \Delta x)
\therefore \Delta y = f(x + \Delta x) - f(x)
Dividing both sides by \Delta x, then
\frac{{\Delta y}}{{\Delta x}} = \frac{{f(x + \Delta x) - f(x)}}{{\Delta x}}
Taking limit of both sides as \Delta x \to 0
\mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}}\mathop { = \lim }\limits_{\Delta x \to 0} \frac{{f(x + \Delta x) - f(x)}}{{\Delta x}}

Thus, if y is the function of x, then \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f(x + \Delta x) - f(x)}}{{\Delta x}} is called the derivative or the differential coefficient of the function or the derivative of f(x) with respect to x and is denoted by f'(x), y', Dy or \frac{{dy}}{{dx}}.

Note: It may be noted that the derivative of the function f(x) with respect to the variable x is the function f' whose value at x is f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h} provided the limit exists, is called the derivative where h is the increment.

Below we list the notations for derivative of y = f(x) used by different mathematicians in a table.

Name of Mathematician
Notation for Derivative
\frac{{dy}}{{dx}}or\frac{{df}}{{dx}} y' f'(x) Df(x)


I. The student should observe the difference between \frac{{\Delta y}}{{\Delta x}} and  \frac{{dy}}{{dx}}. Where \frac{{\Delta y}}{{\Delta x}} is the quotient of the increment of y and x that is its numerator and denominator can be separated, but \frac{{dy}}{{dx}} is a single symbol for the limiting value of the fraction  \frac{{\Delta y}}{{\Delta x}} when \Delta y and \Delta x can be separated.
II.  \frac{d}{{dx}} attached to any function meaning its differential coefficient with respect to x.
III. The phrase “with respect to” will often be abbreviated into w.r.t



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
y = f(x)
1. Change x to x + \Delta x and y to y + \Delta y
i.e. y + \Delta y = f(x + \Delta x)
2. Find \Delta y by subtraction
i.e. \Delta y = f(x + \Delta x) - y
\Delta y = f(x + \Delta x) - f(x)
3. Divide both sides by \Delta x
i.e. \frac{{\Delta y}}{{\Delta x}} = \frac{{f(x + \Delta x) - f(x)}}{{\Delta x}}
4. Find the limit of \frac{{\Delta y}}{{\Delta x}} where \Delta x \to 0
i.e. \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}}\mathop { = \lim }\limits_{\Delta x \to 0} \frac{{f(x + \Delta x) - f(x)}}{{\Delta x}}

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”.