Derivative or Differential Coefficient of a Function.

Differential calculus or the concept of Derivative and Differential Coefficient was discovered by Isaac Newton (1642-1727) and Gottfried Wilhelm Leibnitz (1646-1716) in the process of solving two old problems one of finding slope of tangent drawn to a curve and another of finding instantaneous velocity of an object in non-uniform motion.

Derivative:

When a variable “y” is defined as a function of another variable “x” or,

f(x)=y

Then , The Derivative or Differential Coefficient of the function “f” at a point “x” or with respect to “x” is the limiting value of:

$\displaystyle\lim_{x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

The derivative of a function of “x” with respect to “x” is denoted by:

$\dfrac{df(x)}{d(x)}$

for example:

If “y” is a function of “x” or f(x)=y whose graph looks like:

Then the derivative of the function “f” with respect to “x” at point “x” is :-

$\dfrac{df(x)}{d(x)}=\displaystyle\lim_{x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}=\displaystyle\lim_{x\to 0}\frac{\Delta y}{\Delta x}$

which can be shown in figure as:

Basic properties or theorems of limit. Four basic properties of limits of a function or limit...
Right hand and Left hand limit of a function. The limit of a function have two categories : Left...
The Logarithmic Function. There are many ways to describe logarithmic function. One basic...
Continuity of a function(continuous and discontinuous functions). Continuity of a function , a function can either be...
The concept of Limit. Limit of a function is the value to which function...