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DifferentiationLessonVideoProblems(Sem2)

1) BasicChainRulewithPowerFindthederivativeof 2)23( += xy

2) BasicChainRulewithSquareRoot

Findthederivativeof xy 46 -=

3) BasicChainRulewithFraction

Findthederivativeof136

22 -+

=xx

y

4) Using

Finddxdy intermsofywhen 52 4 -= yx

5) BasicChainRuleApplication

Findthecoordinatesofthestationarypointson 42 )4( -= xy

6) ExponentialDerivatives

Findthederivativeof xey = and xey 53 --= and3

23xey =

7) ExponentialDerivativeswithSimplificationFirst

Findthederivativeof 3xey = andx

x

eey

12 +

=

8) ExponentialDerivativeApplication

Findanequationofthenormaltothecurve 23 -= xey atthepoint ),1( e .

dxdy

dydx=

/1

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9) NaturalLogDerivativesFindthederivativeof xy ln= and xy 6ln= and xy ln7-=

10) NaturalLogDerivativeswithSimplificationFirst

Findthederivativeof 4ln xy = and )ln( 4xey = and ÷øö

çèæ

-+

=5213ln

xxy

11) NaturalLogDerivativeApplicationFindthecoordinatesofthepointofintersectionof )13ln( -= xy and

)2ln( += xy .Thenfindthegradientofeachcurveatthepointofintersection.

12) ProductRulewithoutChainRuleFindthederivativeof xxy ln3= and xexy 2=

13) ProductRulewithChainRule

Findthederivativeof xexy 234= and 23 )15( += xxy

14) ProductRuleApplicationFindtheminimumvalueof xxxf ln)( 3=

15) QuotientRulewithoutChainRule

Findthederivativeofxxy

-+

=551 and

2lnxxy =

16) QuotientRulewithChainRule

Findthederivativeof2)21( x

xy+

= and34

12

-=

+

xeyx

17) QuotientRuleApplication

Showthatifxxy4121

-+

= then32

2

)41( xk

dxyd

-= forkconstant,andfindthevalue

ofk.

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18) BasicDerivativesofSinandCoswithoutChainRuleFindthederivativeof xy sin= and xy cos=

19) BasicDerivativesofSinandCoswithChainRuleFindthederivativeof )5sin( xy = and )13cos(2 -= xy

20) BasicDerivativesofSinandCoswithPowersandChainRuleFindthederivativeof xy 4cos2-= and xy sin=

21) DerivativesofSinandCoswithotherfunctionsandChainRuleTwiceFindthederivativeof )3sin( xey = and )ln(cos2 xy =

22) DerivativesofSinandCosApplicationAparticleismovingalongastraightlineanditsdisplacement,xmetres,fromafixedpointOonthelineattimetseconds,isgivenby ttx sin3cos4 += .

i. Findthevelocityoftheparticlewhen4p

=t .

ii. Findthedisplacementwhenthevelocityisfirstzero.

iii. Findtheaccelerationwhen3p

=t .

23) DerivativesofSinandCoswithProductandQuotientRuleswithoutChainRule

Findthederivativeof xxy sin2= andxexy cos

=

24) DerivativeofSinandCoswithProductRuleFindthederivativeof xxy cossin2=

25) DerivativeofSinandCoswithQuotientRuleandChainRule

Findthederivativeofxxy4cos4sin

=

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26) BasicDerivativesofotherTrigfunctionswithoutChainRuleFindthederivativeof xy tan= and xy cot= and ecxy cos= and xy sec=

27) BasicDerivativesofotherTrigfunctionswithChainRuleFindthederivativeof )5sec( xy = and xy 2tan=

28) DerivativesofotherTrigfunctionswithChainRuleTwiceFindthederivativeof )4ln(tan xy =

29) Usingdxdy

dydx=

/1 withotherTrigfunctionsandtheChainRule

Finddxdy intermsofywhen yx 3cot

61

-=

30) DerivativeofotherTrigfunctionsApplicationFindanequationofthetangenttothecurve xxy tansec2 -= atx=0.

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Answers:

1) 18𝑥 + 122) − '

()*+

3) )'*+)(

(6𝑥2+3𝑥−1)2

4) 01𝑦3

5) 0, 256 ; 2, 0 ; (−2, 0)6) 𝑒𝑥; 15𝑒−5𝑥; 2𝑥𝑒3𝑥2 7) 𝑒3;𝑒𝑥+18) 𝑦 − 𝑒 = − 0

:;(𝑥 − 1)

9) 0+; 0

+; − <

+

10) 0*+; 4; :

:+>0− '

'+)?

11) (:', 𝑙𝑛 <

')

12) 3 1 + 𝑙𝑛𝑥 ; 𝑥𝑒+(𝑥 + 2)13) 4𝑥2𝑒2𝑥 2𝑥 + 3 ;𝑥2(5𝑥 + 1)(25𝑥 + 3)

14) (𝑒−13, − 0

:;)

15) '((5−𝑥)2

; 0)𝑙𝑛𝑥2

𝑥3

16) 0)'+(1+2𝑥)3

; '𝑒2𝑥+1(*+)?)(4𝑥−3)2

17) 𝑘 = 4818) 𝑐𝑜𝑠𝑥;−𝑠𝑖𝑛𝑥19) 5𝑐𝑜𝑠 5𝑥 ;−6𝑠𝑖𝑛(3𝑥 − 1)20) 8𝑠𝑖𝑛𝑥𝑐𝑜𝑠3𝑥; GHI+

' IJK+

21) 3𝑐𝑜𝑠 3𝑥 𝑒sin 3𝑥 ; −2𝑡𝑎𝑛𝑥

22) ) ''𝑚𝑠−1; 5𝑚;−2 − : :

'𝑚𝑠−2

23) 𝑥 𝑥𝑐𝑜𝑠𝑥 + 2𝑠𝑖𝑛𝑥 ; IJK+>GHI+);R

24) 2𝑐𝑜𝑠(2𝑥)25) 4𝑠𝑒𝑐24𝑥26) 𝑠𝑒𝑐2𝑥;−𝑐𝑜𝑠𝑒𝑐2𝑥;−𝑐𝑜𝑡𝑥𝑐𝑜𝑠𝑒𝑐𝑥; 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥27) 5𝑠𝑒𝑐 5𝑥 𝑡𝑎𝑛 5𝑥 ; 2𝑡𝑎𝑛𝑥𝑠𝑒𝑐2𝑥28) 8𝑐𝑜𝑠𝑒𝑐 8𝑥 29) 2𝑠𝑖𝑛2(3𝑦)30) 𝑦 − 2 = −1(𝑥 − 0)