Differentiation from First Principles (Edexcel AS Maths): Revision Note

First Principles Differentiation

What is the derivative or gradient function?

• For a curve y = f(x) there is an associated function called the derivative or gradient function

• The derivative of f(x) is written as f'(x) or

• The derivative is a formula that can be used to find the gradient of y = f(x) at any point, by substituting the x coordinate of the point into the formula

• The process of finding the derivative of a function is called differentiation

• We differentiate a function to find its derivative 

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