Roberto s Notes on Differential Calculus Capter 5: Derivatives of transcendental functions Section Derivatives of Logaritmic functions Wat ou need to know alread: Definition of derivative and all basic differentiation rules. Wat ou can learn ere: How to differentiation a logaritmic function. In te section on inverse functions I included, as an eample, te formula for te derivative of te natural logaritm. Here it is for our convenience: Proof If Tecnical fact: Te natural logaritm rule ln, ten ' B using te metod of implicit differentiation, we start from our function and appl its inverse, te eponential, to bot sides: ln e Net we differentiate implicitl bot sides and isolate : But we ave just seen tat as claimed. e ' ' e e Eample: ln ln, so tat we can conclude tat: ln Te presence of te natural logaritm suggests te use of te corresponding rule, but we need to also use te cain rule, since tis is a composite function: ln ln We can now use appropriate rules to differentiate inside te bracket, including using te one for te natural logaritm again, tus concluding tat: ln Differential Calculus Capter 5: Derivatives of transcendental functions Section : Derivatives of logaritmic functions Page
Te formula for te derivative of te natural logaritm can be easil etended to a formula for te derivative of an logaritmic function. Proof If Tecnical fact: Te general logaritm rule log a, wit a 0, ten B using te cange of base formula, we see tat: ln loga ln a ' ln a But ln a is a constant, so we can combine te logaritm rule and te coefficient rule to obtain: ' ln ln a ln a f log 3ln0 log log 3 ln If ou prefer, we can avoid using different formulae for different logaritm and start b canging all to natural logaritms: f ln 3 log 3 ln0 ln ln 3 log ln ln0 ln ln Te first fraction is now a constant coefficient tat does not affect te derivative, so we focus on te remaining fraction wit te natural logaritm rule and oters as needed: f ln ln 3 ln ln ln 3 ln0 ln ln 3 ln ln0 ln ln 3 Eample: f log 3 log Since tis function consists of a quotient, we start wit te quotient rule. Witin it we ten use te general logaritm rule and oter suitable ones: f log 3 log log log 3 log Wic of te two approaces sown in tis eample is better? Tat is up to ou, depending on wic metod makes more sense and is clearer to ou. As in all oter similar situations, an valid metod is acceptable; just don t use an invalid one! I conclude tis sort section wit te proof of an interesting fact tat ou ma ave seen previousl. Differential Calculus Capter 5: Derivatives of transcendental functions Section : Derivatives of logaritmic functions Page
lim e 0 lim Tecnical facts e Proof Now we know tat te derivative of f ( ) ln is f '( ). Terefore: f ( ) f () ln( ) ln f '() lim lim 0 0 B using basic properties of te natural logaritm, we can see tat: ln( ) lim lim ln( ) 0 0 If we take eponentials on bot sides and use te continuit of te eponential function, we obtain: If, in tis formula, we use On te oter and, if we let limln( ) 0 ln( ) e e e lim lim( ) 0 0 we obtain te first limit: elim( 0 ) we obtain te second: e lim B te wa, te latter limit reflects te wa in wic, istoricall, te number e was discovered. It came from te world of finances and it is an interesting stor, but beond our current goals. Summar Te derivative of an logaritmic function can be obtained b considering it as te inverse of te corresponding eponential, or b using te cange of base formula. Common errors to avoid Remember tat te simple rule onl applies to te natural logaritm: do not appl it in te same wa to oter logaritms. Te logaritm is a single function, so wen it is combined wit oter functions oter appropriate rules ma also appl, suc as product, cain etc. Differential Calculus Capter 5: Derivatives of transcendental functions Section : Derivatives of logaritmic functions Page 3
Learning questions for Section D 5- Review questions:. Describe ow to differentiate a function tat involves a logaritm. Memor questions:. Wat is te derivative of te function ln?. Wat is te derivative of te function log a? 3. Wat is te value of lim? Compute te derivatives of te functions presented in questions -. Computation questions:. 3 ln. ln 3. ln 3 e 3 4. ln e 5. log 3 6. log e 7. 0 log 0 5 3 8. log ln 9. ln 0. ln e. 3 5 e. ln ln e Differential Calculus Capter 5: Derivatives of transcendental functions Section : Derivatives of logaritmic functions Page 4
3. Compute te second derivative of te function ln 4. Compute te second derivative of te function. ln. d 5. Compute ln 3 3ln 5. d Notice tat tere are two was to answer tis question: bot are acceptable. if is defined b te equation Teor questions:. Wat metod is use to obtain te derivative of te natural logaritm?. Wat is te derivative of ln k, for an positive number k? Proof questions: d d. Sow tat ln b using te fact tat.. Sow tat te general logaritm rule applies also to te natural logaritm, in wic case it reduces to te natural logaritm rule. Application questions: 4 at te point, 3 8. Wat is te equation of te line tangent to te curve 4 ln e 3? Templated questions:. Construct a function tat involves a logaritm and compute its derivative Differential Calculus Capter 5: Derivatives of transcendental functions Section : Derivatives of logaritmic functions Page 5
Wat questions do ou ave for our instructor? Differential Calculus Capter 5: Derivatives of transcendental functions Section : Derivatives of logaritmic functions Page 6