Lesson#23and24PowerRule
Differentiation:TheprocessoffindingaderivativeWhatisthederivativeofaconstantfunction?Why?PowerRule: !
!"𝑥! = 𝑛𝑥!!!
Ø “n”isanyrealnumber
Example#1:
𝑓 𝑥 = 𝑥!
Example#2:
ℎ 𝑥 = 𝑥!!
PowerRulewithaConstant: !
!"𝑐𝑥! = 𝑐𝑛𝑥!!!
Example#3:
𝑓 𝑥 = 3𝑥!Example#4: ℎ 𝑥 = 2 𝑥Example#5: 𝑔 𝑥 = −4𝑥! + 2𝑥
!!
Rule:Example#6:
𝑚 𝑥 = 2𝑥! + 3𝑥 − 24Atwhatvalueofxwillthetangentlinetom(x)haveaslopeof7?Findtheequationofthetangentlineandthenormallinetom(x)atx=2.
Lesson#23and24Example#7:Canwefindthederivativeusingpowerrule?
𝑦 =𝑥! − 4𝑥
𝑥
Example#8:Canwefindthederivativeusingpowerrule?
𝑦 =𝑥! − 4𝑥𝑥 + 1
YouTry:1)Findy’andy’’.
𝑦 =2𝑥! − 4
𝑥 2A)Find𝑓′(𝑥), 𝑓 ′′(𝑥), 𝑓′′′(𝑥), 𝑓!"(𝑥).𝑓(𝑥) = (3𝑥 − 2)!
B)Atwhatvalueofxwillthetangentlinetof(x)haveaslopeof1?C)Findtheequationofthetangentlineandthenormallineatx=1.
Lesson#23and24ProductandQuotientRule
ProductRule:Example#1:
ℎ 𝑥 = 6𝑥! − 2 2𝑥! + 𝑥 Example#2:
ℎ 𝑥 = ( 3𝑥 − 2 )!QuotientRule:Example#3:Find!"
!"when𝑦 = !!!!
!!!
_________________________________________________________________________________________________________Notation:!
!!!"!isanotherwayofwriting“secondderivative”
!"!" !!!
meansplugin1,afteryoutakethederivative.Sameas𝑓’(1)_________________________________________________________________________________________________________Example#4:
𝑑𝑑𝑥
[ 𝑓(𝑥)𝑔(𝑥) ] = 𝑓(𝑥)𝑔!(𝑥) + 𝑔(𝑥)𝑓′(𝑥)
𝑑𝑑𝑥
!𝑓(𝑥)𝑔(𝑥)
! =𝑔(𝑥)𝑓!(𝑥) − 𝑓(𝑥)𝑔′(𝑥)
(𝑔(𝑥))!
Lesson#23and24Example#5:Giventhat𝑓 2 = 3, 𝑓! 2 = 1,𝑔 2 = 4,𝑎𝑛𝑑 𝑔! 2 = −3.Find𝐹! 2 𝑤ℎ𝑒𝑛 𝐹 𝑥 = !(!)
!(!)