Limits & First Principles

 

 

 

Contents

   

Limits

  

First Principles

  

Calculus Main Page

Differentiation by First Principles

 
 
Differentiation is one of the two fundamental processes in calculus, the other being integration. In our section on the history of calculus, we have mentioned that calculus came about rather differently than most other mathematical concepts: applications were formulated first, followed by general formulas and geometric representation, and finally a rigorous proof.
 
Earlier mathematicians like Newton and Leibniz has attempted to show the derivation of the concept of differentiation, but the widely accepted one is by d'Alembert, which involve the use of limits.
 
First Principles
We shall formally define the gradient of a curve: the gradient of the tangent to the curve at a particular point, the tangent being the line which cuts the graph at exactly one point.
 
Given a curve y = f(x) as below, with points A (a, f(a)) and B (b, f(b)):
                                                         
 
Consider the gradient of the chord AB:
                                                
 
As B moves towards A (tends towards A), the gradient becomes
                                                
 
Since b = a + dx and dx is tending to zero,
                                               
 
The limit symbol can be abbreviated:
                                               
 
dy/dx can also be written as f'(x) or d/dx [f(x)]
 
 
Example: Using the first principles method, find the derivative of x2 + 2x.
 
Solution:
 
When x changes by dx,
               dy = (x + dx)2 + 2(x + dx) - (x2 - 2x)