ECON 2010 - MATF_Z01.qxd 18:07 Page 645 APPENDIX 1...

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APPENDIX 1 Differentiation from First Principles We hinted in Section 4.1 that there was a formal way of actually proving the formulae for derivatives. This is known as ‘differentiation from Frst principles’ and we begin by illustrating the basic idea using a simple example. ±igure A1.1 shows the graph of the square function f ( x ) = x 2 near x = 3 . The slope of the chord joining points A and B is = = == 6 x Now, as we pointed out in Section 4.1, the slope of the tangent at x = 3 is the limit of the slope of the chords as the width, Δ x , gets smaller and smaller. In this case slope of tangent = lim Δ x 0 (6 x ) = 6 In other words, the derivative of f ( x ) = x 2 at x = 3 is 6 (which agrees with f ( x ) = 2 x evaluated at x = 3 ). Notice that this proof is not restricted to positive values of Δ x . The chords in ±igure A1.1 could equally well have been drawn to the left of x = 3 . In both cases the slope of the chords approaches that of the tangent at x = 3 as the width of the interval shrinks.
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ECON 2010 - MATF_Z01.qxd 18:07 Page 645 APPENDIX 1...

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