LECTURE: 36DERIVATIVES OF LOGARITHMIC FUNCTIONS Review: Derivatives of Exponential Functions: d ex dx ex d dx ax Ma a Example 1: Find a formula for the derivatives of the following functions (a) y lnx (b) y log b x EY em BY b "9b EY by e " IT, 1 by, lnbfhf IF Fx FE Derivatives of Logarithmic Functions: d @ dx ln x d dx log b x TtYyY I Example 2: Find derivatives of the following functions (a) y ln(4x 2 + 5) (b) y ln(tan x) Y 4*5 off (4 45) IaEI@IIsEyax UAF Calculus I 1 36 Derivatives of Logarithmic Functions
( Example 3: Find derivatives of the following functions (a) f(x) log 10 p x (b) g(x) log2 (cos x) f lx) Hoax nl#tfxkgtlx)1tsinx)inotrtztxh 7 inotrxirtx ;, n@ Example 4: Differentiate f and find the domain of f 0 (a) f(x) p 5+lnx ftxttlstinxs " ( 5tln ) " (b) f(x) x 1 ln(x + 1),± 1 20*1 I at#stlnx)tyfmlywylfxtyx ax#s Example 5: Differentiate the following functions I (a) y ln x (b) f(x) ln sec x + tan x Ttnx > O D:lt,et)ulet,xcax1 ( positive ) then 1 111 Tfnxcolneg and ymhlnxj jy ftxtgexlnanxadxslecxttanx ) cash ) then 1 111 secxlanxtsedx and ) secxctnxse#s0_ifyln1xlj lnlxtlntxl s y t Yx secxtlunx secxttanx xtl xtnlnlxtl )+X Ftse 2tiIYYntttjYY#f need X > 0 and 5tlnX70 xtt xtl > 0 > 1 " " MX7 5 ng also I lncxtl ) # 0 eintyy! et, YY w/noaks!f sec C UAF Calculus I 2 36 Derivatives of Logarithmic Functions
It is often easier to first use the rules of logarithms to expand a logarithmic expression before taking the derivative To do this properly you first must recognize when these rules can be applied and apply them correctly Rules and NonRules for Logarithms ln(ab) ln(a/b) ln(a r ) In A + In B ln(a + B) No rule t 1nA HNB In A In B NO rule ln(a B) t In A r In A (ln A) r No rule # r In A 1h13 Example 6: Differentiate the following functions by first expanding the expressions using the rules for logarithms Explain why this is the better way to proceed in each case (a) f(x) ln p 5x +2 (b) g(x) log 5 (x 2p x + 1) In 15 +442 2 logs X + 3 In 15 +4 9 4) tzlogslxtl ),n}t T 4nl H + ) t n±tt5 en Est±en #Fx no u! x Yndiharewnoatdehpaonnsain Example 7: Differentiate f(x) ln x(x 2 + 1) 2 p 2x 4 5 421545k LIEGE fl )1nX + z n( 2+ ) tz In (2 45) fk " T + ftpaxzkl#)8x3xt+x4*4/2x4 UAF Calculus I 3 36 Derivatives of Logarithmic Functions
Example 8: Differentiate the following functions (a) f(x) (lnx) 5 (b) f(x) lnx ( 5 ) ftx ) 5l1nXYiYxsan@IFxtEgfxIYIxyf esgi Logarithmic Differentiation Finding derivatives of complicated functions involving products, quotients and powers can often be simplified using logarithms This technique is called logarithmic differentiation * Example 9: Find the derivative of y x7p x 3 +1 (5x + 1) 4 10g both sides to expand l my mmfxx n ) my 7 lnx + ztlnlxtl ) need to use c) to find y you Guotient, Pwdrett chain rules! 4 In 15 +1 ) the ytddf, + I,, 3 2 hat, 5 save an d% 1 + III, text, )y inputs tlt#zefareftdlxkoedf : you dont input y, your problem is wrong / incomplete UAF Calculus I 4 36 Derivatives of Logarithmic Functions
In lnllnx Derivative Rules: Let n and a be constants (Note, there is no rule when there is a variable in the base and the exponent) xn ( in a) a d dx xn n d dx ax here the exponent is constant here the base is constant When you have a variable in both the base and the exponent you must use Logarithmic differentiation to find the derivative of the function Example 10: Find the derivatives of the following functions using logarithmic differentiation (a) y x 2/x lny my yt In x " X 2 2 lnx + dd (2 ty _)x4@ Zxitx ty ( 2M + 25,4mL ) y a real s yx4 lny lny (b) y (lnx) cos x In ( In ) Asx oosx lnllnx ) yt sinx ) tvosxantx tx 1 94 sinxlnunx )) y 1,9 sinxlnunx ) ) Into " UAF Calculus I 5 36 Derivatives of Logarithmic Functions
Example 11: Find an equation of the tangent line to f(x) ln(x +lnx) at x 1 f an # Cxtlnx ) xtn ( It Yx ) m, # ( Hk ) m 2 111, y fa ) In ( Hlnl ) IN O ytl#y@fcx)ctg,tngaosx Example 12: Let f(x) cx +ln(sinx) For what value of c is f 0 ( /4) 6? f C x ) C + cos %in 6 c + cosy sin T14 6 c + 1 cg UAF Calculus I 6 36 Derivatives of Logarithmic Functions