First Principles Differentiation (OCR AS Maths: Pure)

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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 fraction numerator d y over denominator d x end fraction
  • 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

space f apostrophe left parenthesis x right parenthesis equals limit as h rightwards arrow 0 of space fraction numerator f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis over denominator h end fraction

  • limit as h rightwards arrow 0 of means the 'limit as h tends to zero'
  • Whenspace h equals 0space fraction numerator f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis over denominator h end fraction equals fraction numerator f left parenthesis x right parenthesis minus f left parenthesis x right parenthesis over denominator 0 end fraction equals 0 over 0 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
   space f left parenthesis x right parenthesis equals 3 x squared so f apostrophe left parenthesis x right parenthesis equals stack l i m with h rightwards arrow 0 below fraction numerator f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis over denominator h end fraction equals stack l i m with h rightwards arrow 0 below fraction numerator 3 left parenthesis x plus h right parenthesis squared minus space 3 x squared over denominator h to the power of blank end fraction

STEP 2: Expand f(x+h) in the numerator
   space f apostrophe left parenthesis x right parenthesis equals stack l i m with h rightwards arrow 0 below fraction numerator 3 left parenthesis x squared plus 2 h x plus h squared right parenthesis minus 3 x squared over denominator h end fraction
   space f apostrophe left parenthesis x right parenthesis equals stack l i m with h rightwards arrow 0 below fraction numerator 3 x squared plus 6 h x plus 3 h squared minus 3 x squared over denominator h end fraction

STEP 3: Simplify the numerator, factorise and cancel h with the denominator
   space f apostrophe left parenthesis x right parenthesis equals stack l i m with h rightwards arrow 0 below fraction numerator h left parenthesis 6 x plus 3 h right parenthesis over denominator h end fraction
STEP 4: Evaluate the remaining expression as h tends to zero
    space f apostrophe left parenthesis x right parenthesis equals stack l i m with h rightwards arrow 0 below left parenthesis 6 x plus 3 h right parenthesis equals 6 x     A s space h rightwards arrow 0 comma space left parenthesis 6 x plus 3 h right parenthesis rightwards arrow left parenthesis 6 x plus 0 right parenthesis rightwards arrow 6 x
thereforeThe derivative of 3 x squared is 6 x

Examiner Tip

  • Most of the time you will not use first principles to find the derivative of a function (there are much quicker ways!). However, you can be asked on the exam to demonstrate differentiation from first principles.
  • Make sure you can use first principles differentiation to find the derivatives of kx, kx2 and kx3 (where k is a constant).

Worked example

1st Princ Diff Example, A Level & AS Maths: Pure revision notes

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.