Differentiatinglogarithm andexponential functionsmc-TY-logexp-2009-1This unit gives details of how logarithmic functions and exponential functions are differentiatedfrom first principles.In order to master the techniques explained here it is vital that you undertake plenty of practiceexercises so that they become second nature.After reading this text, and/or viewing the video tutorial on this topic, you should be able to:• differentiate ln x from first principles• differentiate exContents1. Introduction 22. Differentiation of a function f(x) 23. Differentiation of f(x) = ln x 34. Differentiation of f(x) = ex 4www.mathcentre.ac.uk 1 c mathcentre 20091. IntroductionIn this unit we explain how to differentiate the functions ln x and exfrom first principles.To understand what follows we need to use the result that the exponential constant e is definedas the limit as t tends to zero of (1 + t)1/t i.e. limt→0(1 + t)1/t.To get a feel for why this is so, we have evaluated the expression (1 + t)1/t for a number ofdecreasing values of t as shown in Table 1. Note that as t gets closer to zero, the value of theexpression gets closer to the value of the exponential constant e≈ 2.718…. You should verifysome of the values in the Table, and explore what happens as t reduces further.t (1 + t)1/t1 (1 + 1)1/1 = 20.1 (1 + 0.1)1/0.1 = 2.5940.01 (1 + 0.01)1/0.01 = 2.7050.001 (1.001)1/0.001 = 2.7170.0001 (1.0001)1/0.0001 = 2.718We will also make frequent use of the laws of indices and the laws of logarithms, which shouldbe revised if necessary.

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