First Principles Differentiation (DP IB Analysis & Approaches (AA)): Revision Note

First principles differentiation

What is differentiation from first principles?

•  Differentiation from first principles uses the definition of the derivative of a function f(x)

• The definition is

•  means the 'limit as h tends to zero'

• When,  which is undefined

  • Instead we consider what happens as h gets closer and closer to zero

• Differentiation from first principles means using that definition to show what the derivative of a function is

How do I differentiate from first principles?

• STEP 1
  Identify the function f(x) and substitute this into the first principles formula   

  • E.g.  Show, from first principles, that the derivative of 3x² is 6x

 so 

• STEP 2
  Expand f(x+h) in the numerator 

• STEP 3
  Simplify the numerator, factorise and cancel h with the denominator 

• STEP 4
  Evaluate the remaining expression as h tends to zero

The derivative of  is