Instantaneous rate of Change and Differentiability
When a quantity undergoes change, the change in the quantity is always associated, in almost every field of human endeavor, with the rate at which the quantity change. The rate of change in the quantity may be observed with respect to another quantity such as time, distance, or position in space, etc.
The instantaneous rate of change, i.e. the rate at which the quantity changes at any particular time. If a person travels at a uniform speed. i.e. if there is no change in the rate at which he is changing his position, then the instantaneous rate of change of position at any point would remain a constant.
If is a function whose graph is a straight line then is nearly a constant when is small. i.e. there is a constant on such that , where can be made small by taking close enough to , this means, (Differentiability of at .)
Definition of Derivatives
A function is said to be derivable or differentiable at if exists.
This limit is called the derivative of with respect to (w. r. t) .The derivative of is denoted by or or sometimes or .
We can also write,
Let and be derivable at , then
When (i.e. through negative values only) the limit is called the left- hand derivative of at and is denoted by .
When (i.e. when through positive values only) the limit is called the right hand derivative of at and is denoted by .
It is easily seen that derivative of exists at iff and both exists and are equal and irrespective of sign of .
If has a derivative at every point of the interval , then it is said to be differentiable on . The process of finding the derivative of a function is known as differentiation.