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

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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

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

  • The first principles definition (formula) is in the formula booklet

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     As 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 Tips and Tricks

  • 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 to demonstrate differentiation from first principles

  • To get full marks make sure you are are writing lim h -> 0 right up until the concluding sentence!

Worked Example

Prove, from first principles, that the derivative of  5 x cubed  is  15 x squared.

first-principles-diff-corrected
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Paul

Author: Paul

Expertise: Maths Content Creator (Previous)

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.

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