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
Leibniz
Newton
Lagrange
Cauchy
Notation for Derivative
\frac{{dy}}{{dx}}or\frac{{df}}{{dx}} y' f'(x) Df(x)

Remarks:

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

 

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
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”.