Worksheet: Differentiation of Logarithmic Functions

In this worksheet, we will practice finding the derivatives of logarithmic functions.

Q1:

Find dd๐‘ฆ๐‘ฅ, given that ๐‘ฆ=(4๐‘ฅ+5)ln๏Šญ.

  • A284๐‘ฅ+5
  • B28(4๐‘ฅ+5)
  • C44๐‘ฅ+5
  • D1(4๐‘ฅ+5)๏Šญ

Q2:

Determine dd๐‘ฆ๐‘ฅ, given that ๐‘ฆ๐‘ฆ=๏€ผโˆ’14๐‘ฅโˆ’7๏ˆ:ln๏Šซ.

  • A20๐‘ฅ(4๐‘ฅโˆ’7)๏Šช๏Šซ๏Šจ
  • B20๐‘ฅ4๐‘ฅโˆ’7๏Šช๏Šซ
  • Cโˆ’20๐‘ฅ4๐‘ฅโˆ’7๏Šช๏Šซ
  • Dโˆ’20๐‘ฅ(4๐‘ฅโˆ’7)๏Šช๏Šซ๏Šจ

Q3:

Find dd๐‘ฆ๐‘ฅ, given that ๐‘ฆ=๏€พโˆ’8๐‘ฅ7๐‘ฅโˆ’3๏Šln๏Šจ.

  • A15๐‘ฅโˆ’37๐‘ฅโˆ’3๐‘ฅ๏Šจ
  • B7๐‘ฅโˆ’3๐‘ฅ7๐‘ฅโˆ’6๏Šจ
  • C7๐‘ฅโˆ’67๐‘ฅโˆ’3๐‘ฅ๏Šจ
  • D7๐‘ฅโˆ’3๐‘ฅ15๐‘ฅโˆ’3๏Šจ

Q4:

Find dd๐‘ฆ๐‘ฅ, given that ๐‘ฆ=3๐‘ฅ3๐‘ฅ๏Šฌ๏Šฉln.

  • A9๐‘ฅ+18๐‘ฅ3๐‘ฅ๏Šซ๏Šซ๏Šฉln
  • B9๐‘ฅ+18๐‘ฅ3๐‘ฅ๏Šฉ๏Šซ๏Šฉln
  • C3๐‘ฅ+18๐‘ฅ3๐‘ฅ๏Šซ๏Šซ๏Šฉln
  • D3๐‘ฅ+18๐‘ฅ3๐‘ฅ๏Šฉ๏Šซ๏Šฉln

Q5:

Differentiate ๐น(๐‘ก)=โˆ’4(๐‘ก)2๐‘กlnsin๏Šจ.

  • A๐นโ€ฒ(๐‘ก)=8๐‘ก(๐‘ก๐‘ก2๐‘ก+2๐‘ก)๐‘กlncossinln
  • B๐นโ€ฒ(๐‘ก)=โˆ’8๐‘ก(๐‘ก๐‘ก2๐‘ก+2๐‘ก)๐‘กlncossinln
  • C๐นโ€ฒ(๐‘ก)=8(๐‘ก2๐‘ก+2๐‘ก)๐‘กlncossinln
  • D๐นโ€ฒ(๐‘ก)=8๐‘ก(๐‘ก๐‘ก2๐‘กโˆ’2๐‘ก)๐‘กlncossinln
  • E๐นโ€ฒ(๐‘ก)=โˆ’8(๐‘ก2๐‘ก+2๐‘ก)๐‘กlncossinln

Q6:

Find dd๐‘ฆ๐‘ฅ, given that ๐‘ฆ=9๐‘ฅ9๐‘ฅln.

  • A9(1โˆ’9๐‘ฅ)9๐‘ฅlnln๏Šจ
  • B9(9๐‘ฅโˆ’๐‘ฅ)9๐‘ฅlnln๏Šจ
  • C9(๐‘ฅโˆ’9๐‘ฅ)9๐‘ฅlnln๏Šจ
  • D9(9๐‘ฅโˆ’1)9๐‘ฅlnln๏Šจ

Q7:

If ๐‘“(๐‘ฅ)=3(2๐‘ฅ+4๐‘ฅ)lnln, find ๐‘“โ€ฒ(1).

  • A19
  • B1
  • C3
  • D9
  • E12

Q8:

Differentiate the function ๐ป(๐‘ง)=๏„ž๐‘Žโˆ’๐‘ง๐‘Ž+๐‘งln๏Šจ๏Šจ๏Šจ๏Šจ.

  • A๐ปโ€ฒ(๐‘ง)=โˆ’2๐‘Ž๐‘ง๐‘งโˆ’๐‘Ž๏Šจ๏Šช๏Šช
  • B๐ปโ€ฒ(๐‘ง)=2๐‘Ž๐‘ง๐‘งโˆ’๐‘Ž๏Šจ๏Šช๏Šช
  • C๐ปโ€ฒ(๐‘ง)=2๐‘Ž๐‘ง๐‘งโˆ’๐‘Ž๏Šจ๏Šช๏Šช
  • D๐ปโ€ฒ(๐‘ง)=๐‘งโˆ’๐‘Ž2๐‘Ž๐‘ง๏Šช๏Šช๏Šจ
  • E๐ปโ€ฒ(๐‘ง)=โˆ’2๐‘Ž๐‘ง๐‘งโˆ’๐‘Ž๏Šจ๏Šช๏Šช

Q9:

Differentiate ๐‘”(๐‘ก)=โˆš4๐‘กโˆ’9ln.

  • A๐‘”โ€ฒ(๐‘ก)=โˆ’4๐‘กโˆš4๐‘กโˆ’9ln
  • B๐‘”โ€ฒ(๐‘ก)=2๐‘กโˆš4๐‘กโˆ’9ln
  • C๐‘”โ€ฒ(๐‘ก)=4๐‘กโˆš4๐‘กโˆ’9ln
  • D๐‘”โ€ฒ(๐‘ก)=โˆ’2๐‘กโˆš4๐‘กโˆ’9ln

Q10:

Differentiate ๐‘“(๐‘ฅ)=5๏€บ2๐‘ฅ๏†lnsin๏Šจ.

  • A๐‘“โ€ฒ(๐‘ฅ)=5๐‘ฅcot
  • B๐‘“โ€ฒ(๐‘ฅ)=10๐‘ฅtan
  • C๐‘“โ€ฒ(๐‘ฅ)=10๐‘ฅcot
  • D๐‘“โ€ฒ(๐‘ฅ)=52๐‘ฅtan
  • E๐‘“โ€ฒ(๐‘ฅ)=โˆ’10๐‘ฅcot

Q11:

Differentiate ๐‘“(๐‘ฅ)=5(5๐‘ฅ)sinln.

  • A๐‘“โ€ฒ(๐‘ฅ)=25(5๐‘ฅ)cosln
  • B๐‘“โ€ฒ(๐‘ฅ)=โˆ’25๐‘ฅ(5๐‘ฅ)cosln
  • C๐‘“โ€ฒ(๐‘ฅ)=โˆ’25(5๐‘ฅ)cosln
  • D๐‘“โ€ฒ(๐‘ฅ)=25๐‘ฅ(๐‘ฅ)cosln
  • E๐‘“โ€ฒ(๐‘ฅ)=25๐‘ฅ(5๐‘ฅ)cosln

Q12:

Given that ๐‘“(๐‘ฅ)=๏€บ๐‘ฅ๏†cosln๏Šจ, determine ๐‘“โ€ฒ(1).

Q13:

Differentiate the function ๐‘ฆ=[(๐‘Ž๐‘ฅ+๐‘)]tanln.

  • A๐‘Ž((๐‘Ž๐‘ฅ+๐‘))๐‘Ž๐‘ฅ+๐‘secln๏Šจ
  • Bsecln๏Šจ((๐‘Ž๐‘ฅ+๐‘))
  • Cโˆ’((๐‘Ž๐‘ฅ+๐‘))secln๏Šจ
  • Dโˆ’๐‘Ž((๐‘Ž๐‘ฅ+๐‘))๐‘Ž๐‘ฅ+๐‘secln๏Šจ
  • E(๐‘Ž๐‘ฅ+๐‘)((๐‘Ž๐‘ฅ+๐‘))๐‘Žsecln๏Šจ

Q14:

Given that ๐‘ฆ=โˆ’34(7๐‘ฅ+7๐‘ฅ)lntansec, determine dd๐‘ฆ๐‘ฅ.

  • Aโˆ’347๐‘ฅsec
  • Bโˆ’347๐‘ฅ+47๐‘ฅtansec
  • Cโˆ’21๏€น7๐‘ฅ+7๐‘ฅ๏…7๐‘ฅ47๐‘ฅ+47๐‘ฅtansecsectansec๏Šจ
  • Dโˆ’2147๐‘ฅsec

Q15:

Given that ๐‘ฆ=8((9๐‘ฅ))lnlnln, find dd๐‘ฆ๐‘ฅ.

  • Aโˆ’8๐‘ฅ9๐‘ฅ(9๐‘ฅ)lnlnln
  • Bโˆ’8(9๐‘ฅ)lnln
  • C8๐‘ฅ9๐‘ฅ(9๐‘ฅ)lnlnln
  • D8(9๐‘ฅ)lnln

Q16:

A factoryโ€™s production of ๐‘ฆ units in day ๐‘ก is governed by the relation ๐‘ฆ=400๏€น10โˆ’๐‘’๏…๏Šฑ๏Šฆ๏Ž–๏Šฎ๏. What is the rate of change of production with respect to time on the fifth day?

  • A0.8๐‘’๏Šช
  • B320๐‘’๏Šช
  • Cโˆ’400๐‘’๏Šช
  • D320๐‘’๏Šช

Q17:

Determine the first derivative of ๐‘ฆ=โˆ’๐‘’+2๐‘ฅ๏‘๏Žฃln๏Šฎ.

  • Aโˆ’๐‘’+2๐‘ฅ๏‘๏Žฃ
  • Bโˆ’๐‘’+16๐‘ฅ๏‘๏Žฃ
  • Cโˆ’๐‘’4+2๐‘ฅ๏‘๏Žฃ
  • Dโˆ’๐‘’4+16๐‘ฅ๏‘๏Žฃ

Q18:

Find the first derivative of the function ๐‘ฆ=๏€นโˆ’5๐‘ฅ+2๐‘ฅ๏…ln๏Šช๏Šจ.

  • Aโˆ’20๐‘ฅ+4๐‘ฅ(โˆ’5๐‘ฅ+2๐‘ฅ)๏Šฉ๏Šช๏Šจln
  • Bโˆ’20๐‘ฅ+4๐‘ฅ๏Šฉ
  • C20๐‘ฅโˆ’4๐‘ฅ(5๐‘ฅโˆ’2)๏Šจ๏Šจ
  • D๐‘ฅ๏€น5๐‘ฅโˆ’2๏…20๐‘ฅโˆ’4๏Šจ๏Šจ

Q19:

Find the first derivative of the function ๐‘ฆ=โˆ’7๐‘ฅ6๐‘ฅ๏Šช๏Šชln.

  • Aโˆ’28๐‘ฅ๏€น6๐‘ฅ๏…โˆ’76๏Šฉ๏Šชln
  • Bโˆ’112๐‘ฅ๏Šจ
  • Cโˆ’28๐‘ฅ๏€น๏€น6๐‘ฅ๏…+1๏…๏Šฉ๏Šชln
  • Dโˆ’28๐‘ฅ๏€น๏€น6๐‘ฅ๏…+1๏…๏Šช๏Šชln

Q20:

Find dd๐‘ฆ๐‘ฅ, given that ๐‘ฆ=4๐‘ฅ+34๐‘ฅโˆ’7lnln.

  • Aโˆ’16๐‘ฅ(4๐‘ฅโˆ’7)ln๏Šจ
  • Bโˆ’40๐‘ฅ(4๐‘ฅโˆ’7)ln๏Šจ
  • Cโˆ’10๐‘ฅ(4๐‘ฅโˆ’7)ln๏Šจ
  • Dโˆ’40๐‘ฅ(4๐‘ฅโˆ’7)ln

Q21:

Differentiate ๐‘ฆ=โˆ’5|5๐‘ฅโˆ’5๐‘ฅ+3|ln๏Šฉ.

  • A๐‘ฆโ€ฒ=โˆ’55๐‘ฅโˆ’5๐‘ฅ+3๏Šฉ
  • B๐‘ฆโ€ฒ=โˆ’75๐‘ฅโˆ’25๐‘ฅ5๐‘ฅโˆ’5๐‘ฅ+3๏Šจ๏Šฉ
  • C๐‘ฆโ€ฒ=โˆ’75๐‘ฅโˆ’25|5๐‘ฅโˆ’5๐‘ฅ+3|๏Šจ๏Šฉln
  • D๐‘ฆโ€ฒ=โˆ’25๐‘ฅโˆ’25๐‘ฅ+1515๐‘ฅโˆ’5๏Šฉ๏Šจ
  • E๐‘ฆโ€ฒ=โˆ’75๐‘ฅโˆ’255๐‘ฅโˆ’5๐‘ฅ+3๏Šจ๏Šฉ

Q22:

Differentiate ๐‘“(๐‘ฅ)=โˆ’5โˆš๐‘ฅ+4ln, and determine its domain.

  • A๐‘“โ€ฒ(๐‘ฅ)=โˆ’52๐‘ฅโˆš๐‘ฅ+4ln, ๏”1๐‘’,โˆž๏ˆ๏Šช
  • B๐‘“โ€ฒ(๐‘ฅ)=โˆ’5๐‘ฅ2โˆš๐‘ฅ+4ln, ๏€ผ1๐‘’,โˆž๏ˆ๏Šช
  • C๐‘“โ€ฒ(๐‘ฅ)=โˆ’5๐‘ฅโˆš๐‘ฅ+4ln, ๏”1๐‘’,โˆž๏ˆ๏Šช
  • D๐‘“โ€ฒ(๐‘ฅ)=โˆ’52๐‘ฅโˆš๐‘ฅ+4ln, ๏€ผ1๐‘’,โˆž๏ˆ๏Šช
  • E๐‘“โ€ฒ(๐‘ฅ)=โˆ’5๐‘ฅ2โˆš๐‘ฅ+4ln, ๏”1๐‘’,โˆž๏ˆ๏Šช

Q23:

Differentiate ๐‘“(๐‘ฅ)=โˆ’๏€น๐‘ฅ+4๐‘ฅ๏…ln๏Šจ, and determine its domain.

  • A๐‘“โ€ฒ(๐‘ฅ)=โˆ’1๐‘ฅ+4๐‘ฅ๏Šจ, โ„
  • B๐‘“โ€ฒ(๐‘ฅ)=โˆ’2๐‘ฅ+4๐‘ฅ+4๐‘ฅ๏Šจ, โ„
  • C๐‘“โ€ฒ(๐‘ฅ)=โˆ’1๐‘ฅ+4๐‘ฅ๏Šจ, (โˆ’โˆž,โˆ’4)โˆช(0,โˆž)
  • D๐‘“โ€ฒ(๐‘ฅ)=โˆ’๐‘ฅโˆ’4๐‘ฅ+4๐‘ฅ๏Šจ๏Šจ, (โˆ’โˆž,โˆ’4)โˆช(0,โˆž)
  • E๐‘“โ€ฒ(๐‘ฅ)=โˆ’2๐‘ฅ+4๐‘ฅ+4๐‘ฅ๏Šจ, (โˆ’โˆž,โˆ’4)โˆช(0,โˆž)

Q24:

Differentiate ๐‘“(๐‘ฅ)=โˆ’4๐‘ฅ3(๐‘ฅโˆ’3)โˆ’3ln, and determine its domain.

  • A๐‘“โ€ฒ(๐‘ฅ)=โˆ’4๐‘ฅ(๐‘ฅโˆ’3)โˆ’12(๐‘ฅโˆ’3)+123(๐‘ฅโˆ’3)((๐‘ฅโˆ’3)โˆ’1)lnlnln๏Šจ, (3,๐‘’+3)โˆช(๐‘’+3,โˆž)
  • B๐‘“โ€ฒ(๐‘ฅ)=โˆ’4๐‘ฅ(๐‘ฅโˆ’3)โˆ’8๐‘ฅโˆ’12(๐‘ฅโˆ’3)+123(๐‘ฅโˆ’3)((๐‘ฅโˆ’3)โˆ’1)lnlnln๏Šจ, (3,๐‘’+3)โˆช(๐‘’+3,โˆž)
  • C๐‘“โ€ฒ(๐‘ฅ)=4๐‘ฅ(๐‘ฅโˆ’3)โˆ’8๐‘ฅโˆ’12(๐‘ฅโˆ’3)+12(๐‘ฅโˆ’3)((๐‘ฅโˆ’3)โˆ’1)lnlnln, (3,๐‘’+3)โˆช(๐‘’+3,โˆž)
  • D๐‘“โ€ฒ(๐‘ฅ)=โˆ’4๐‘ฅ(๐‘ฅโˆ’3)โˆ’8๐‘ฅโˆ’12(๐‘ฅโˆ’3)+123(๐‘ฅโˆ’3)((๐‘ฅโˆ’3)โˆ’1)lnlnln๏Šจ, (โˆ’โˆž,3)โˆช(3,๐‘’+3)โˆช(๐‘’+3,โˆž)
  • E๐‘“โ€ฒ(๐‘ฅ)=โˆ’4๐‘ฅ(๐‘ฅโˆ’3)โˆ’12(๐‘ฅโˆ’3)+123(๐‘ฅโˆ’3)((๐‘ฅโˆ’3)โˆ’1)lnlnln๏Šจ, (โˆ’โˆž,3)โˆช(3,๐‘’+3)โˆช(๐‘’+3,โˆž)

Q25:

Differentiate ๐‘ƒ(๐‘ฃ)=2๐‘ฃ5๐‘ฃโˆ’3ln.

  • A๐‘ƒโ€ฒ(๐‘ฃ)=โˆ’10๐‘ฃ๐‘ฃ+10๐‘ฃโˆ’6(5๐‘ฃโˆ’3)ln๏Šจ
  • B๐‘ƒโ€ฒ(๐‘ฃ)=23๐‘ฃ
  • C๐‘ƒโ€ฒ(๐‘ฃ)=โˆ’10๐‘ฃ๐‘ฃโˆ’10๐‘ฃโˆ’6๐‘ฃ(5๐‘ฃโˆ’3)ln๏Šจ
  • D๐‘ƒโ€ฒ(๐‘ฃ)=โˆ’10๐‘ฃ๐‘ฃ+10๐‘ฃโˆ’6๐‘ฃ(5๐‘ฃโˆ’3)ln๏Šจ
  • E๐‘ƒโ€ฒ(๐‘ฃ)=โˆ’10๐‘ฃ๐‘ฃ+10๐‘ฃโˆ’6๐‘ฃ(5๐‘ฃโˆ’3)ln

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