First Principle of Differentiation

Calculus - Differentiation (Module 1)



Introduction to Differentiation:


(A) The First Principle of Differentiation
The derivative of a function is defined as a limit of a difference quotient as the
variable h approaches zero, as follows:
� �+ℎ −� �
�' � = lim
ℎ→0 ℎ
When we are finding the derivative of a function at a specific point � = � , we can
evaluate the following limit:
� � −� �
�' � = lim
�→� �−�


In the following sections, we will derive the fundamental rules of differentiation by
using the first principle of differentiation.

, Calculus - Differentiation (Module 1)



(A1) Power Rule of Differentiation
� �
� = ���−1
��
Proof:
Let � � = �� . Then, we have:
( x  h) n  x n
f ' ( x)  lim
h 0 h
n
 n  nk k
   x h  x n
k 0 k
 lim  
h 0 h
n
n
x n     x n  k h k  x n
k 1  k 
 lim
h 0 h
n
 n  nk k
   x h
k 1  k 
 lim
h 0 h
n
n
 lim    x n  k h k 1
k 1  k 
h 0


 n  n
n 
 lim   x n 1     x n  k h k 1 
 1  k 2  k 
h 0

 n
n 
 lim nx n 1     x n  k h k 1 
k 2  k 
h 0
 
n 1
 nx


Note: each term of the series becomes zero because they consist of h raised to the
power of at least 1 and h→0.


Extra Information:
Notice that differentiation is a linear operation that satisfies the following properties:
� � �
� � +� � = � � + � �
�� �� ��
� �
�� � = � � �
�� ��