1 The Derivative of ln(x)

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1 Monay, December 3, 2007 The Derivative of ln() 1 The Derivative of ln() The first term or semester of most calculus courses will inclue the it efinition of the erivative an will work out, long han, a number of specific eamples to illustrate more general rules for quickly eveloping erivatives. For polynomials, these rules are relatively easily evelope an applie by stuents. Even in the case of certain transcenental functions, such as sine an cosine where the infinite series can be manipulate an its results then compare, stuents can be quickly shown why it is that: sin cos (1) A simple inspection of the each series an taking the erivative, term by term, is often sufficient to get the point across without too much ifficulty. Of course, that emonstration assumes that the infinite series is grante by the stuent, to start with. An although Taylor's can be brought in, eplaine in principle, an use to evelop the series, the straight-forwar application of Taylor's assumes the erivatives are known beforehan so that the constants of consecutive terms in the Taylor's epansion can be compute. Luckily, all that can be avoie. Trigonometric ientities an a little imaginative use of conjugates can be applie to the it efinition for the erivative to make the case without resorting to the infinite series. In other wors, a stuent's earlier trigonometric founations are sufficient to allow them to follow a clear argument in this case. So, if presse, a teacher can lea stuents through the etails using their prior training an fresh knowlege on eveloping results for it problems. However, this isn't so easy in the case of showing: ln h ln ln 1 (2) h A stuent may be left with accepting the eclaration without seeming to have any way of confirming the result. But that is appearances. The emonstration can be mae using knowlege they probably alreay have prior to taking the first term or semester of calculus together with the early founations being lai in that class. But to get there we nee to consier an interesting it problem. Contents 1 The Derivative of ln() Two Curious Questions Interest rates Probability Two Curious Questions Answere Interest rates Probability Returning to the Derivative of ln() Summary

2 Two Curious Questions Monay, December 3, Two Curious Questions There is a fascinating it problem that evolves from a very practical question namely, effective interest rates as well as a curious probability question. 2.1 Interest rates Interest can be compoune meaning that not only is interest pai on the principle amount but also that interest is also pai on earlier accumulate interest, as well. For eample, a 10% per annum might be specifie, but interest might be accrue an then consiere a part of the principle for further calculations, more often than once a year. Let's say the number of equal perios use in a year is n. To compute the effective interest rate over the perio, the annual interest rate is equally ivie an then applie, yieling the following equation: r e 1 r n a n 1, where r e is the effective rate an r a is the simple rate (3) But what if we compoun this constantly? In other wors, what happens as n? 2.2 Probability My son aske himself a simple question. What if the os of something happening were fairly unlikely, but we epane the number of trials to compensate for that? In other wors, what if we increase the number of trials in inverse proportion to the os? What chance woul there still be for a failure, then? Perform 6 trials when there is only 1 chance in 6? Or perform 100 trials when there is only 1 chance in 100? Etc. He wante to know what the os for not one success might be, if more trials were performe for less likely events. He was able to epress the basic iea from his knowlege of combining probabilities of inepenent events. The probability, between 0 an 1, is: P failure 1 1, given each inepenent event succees once in trials (4) The question he then aske was, what happens as the os ecline arbitrarily an the number of trials increase, proportionately, as? Both of these questions can be answere by eploring this it: n 1 a n, where a is some constant we can choose n (5) This is where we will now focus our attention. 2

3 Monay, December 3, 2007 Two Curious Questions Answere 3 Two Curious Questions Answere Let's take a few initial steps along the way an then focus on a familiar portion: 1 a ] a ] a ] (6) Epaning the numerator portion of the right sie in (6) provies some iea about how we might epress this in a slightly ifferent form: the first five epansions, starting with 0 an proceeing to 5. I'll leave in the epane terms for familiarity an placeholer purposes, but keep in min that it has a ifferent numeric value in each case: a a 1 a 1 a a a 2 2 a a 2 3 a 2 a 3 3 a a 3 6 a a 3 a 4 4 a a 4 10 a a a 4 a 5 5 This is a familiar binomial series epansion. The constants in each term come from Pascal's triangle an the general form of the binomial series ientity is: a n n k 0 n k n k a k ], where the binomial coefficient n k n! k! k! We can then substitute the summation in (8) into the right-sie epression in (6) to get: a k]! ] k0 k! k! k a 1 k]! k 0 k! k! k a The term 1 oesn't vary on k, so we can move it into the summation with the plan of then combining it with the term, k 0 k, foun insie the summation epression: ak] k]! k! k! k a (10)! k! k! k I'm going to re-arrange the terms in (10) a little bit, so that we can focus on an interesting part of the epression: k 0! k! k 0 (7) (8) (9) ak k k!] (11) 3

4 Two Curious Questions Answere Monay, December 3, 2007 Let's look at the first factor in the summation term, consiering ifferent values of k.! 0! 0! 2!!! 1!!! 1!!!! 1! 1 1, where k0 1, where k1 1, where k 2 (12) k 1 j j0 1, k 0 As, the term remains 1 for any finite k. To summarize:! k 0 k! ] k ak k! ak k0 1 a ] k!] The infinite series inicate in the latter part of (13) is a well-recognize epansion. We can now answer the earlier two questions. 3.1 Interest rates The effective interest rate, compoune constantly, can now be complete as: r e n 1 r a n n ] 1 k0 ak k!] ea (13) e r a 1 (14) An interest rate of 8% per annum, compoune constantly, becomes an effective rate of about 8.33%. In other wors, $100 woul yiel $8.33 in one year, rather than just the $8 that woul have otherwise happene if simple interest were being applie, rather than being compoune constantly. 3.2 Probability A similar answer applies to the probability question, as the event likelihoo iminishes but also the number of trials increases. The result looks like: P failure 1 1 ] 1 1 Now on towars solving the erivative of ln(). ] e (15) 4

5 Monay, December 3, 2007 Returning to the Derivative of ln() 4 Returning to the Derivative of ln() A really interesting question comes from wonering how it might be that we can fin the erivative of a transcenental equation to be a simple rational fraction! How is it that: ln (16) Epresse slightly ifferently an using the it theorem for fining the erivative, we have: ln h ln 1 h 0 h From our knowlege about subtraction of logarithms, we can rephrase this as: ln h ln ln h 0 h 1 h ln h ] Of course, that oesn't help a lot. But the solution is near to han. Let's rephrase (18) this way: ln (17) 1 h ln 1 h ] (18) The term, 1 h ln 1 h ] (19) 1, can be move outsie the it since it oesn't vary as h 0. 1 h 0 h ln 1 h ] (20) We can now procee to apply another property when multiplying a logarithm by a term: 1 h 0 { ln 1 h h}] (21) Given our earlier work, we can almost see the answer. Let's create a new variable, efine as n h an apply it to our formula. Keep in min that as h 0, n. 1 ln n { 1 1 n n }] (22) Of course, we now know what happens to that epression insie the logarithm! It becomes the value of e. It is easy to see where this is heae, now: ln 1 n { ln 1 1 n n }] 1 ln e ] 1 n 1 1 n (23) 5

6 Summary Monay, December 3, Summary Both the fun, perhaps as well as one of those necessary facts of oing a little mathematics, is that sometimes an ecursion into seemingly unrelate areas may provie a tool neee at a ifferent time an where, without having ha that earlier eposure, the approach might not have been so easily recognize. The trip might be stimulate by a practical question, such as the one about interest rates, or else by way of a hypothetical probability question born more out of moest curiosity. Or something else, still. But searching an fining that answer provie a useful tool to help fin the erivative for an interesting transcenental function. Serenipity is a familiar face in mathematics. 6