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 = !(!)

!(!)