Worksheet: Logarithmic Differentiation

In this worksheet, we will practice finding the derivatives of positive functions by taking the natural logarithm of both sides before differentiating.

Q1:

Using logarithmic differentiation, determine the derivative of ๐‘ฆ=๏„ž๐‘ฅ+12๐‘ฅโˆ’2๏Šช.

  • A๐‘ฆโ€ฒ=๏„ž๐‘ฅ+12๐‘ฅโˆ’2๏€พ12๐‘ฅ+2โˆ’2๐‘ฅ๐‘ฅโˆ’1๏Š๏Šช๏Šฉ๏Šช
  • B๐‘ฆโ€ฒ=๏„ž๐‘ฅ+12๐‘ฅโˆ’2๏€พ12๐‘ฅ+2โˆ’๐‘ฅ2๐‘ฅโˆ’2๏Š๏Šช๏Šฉ๏Šช
  • C๐‘ฆโ€ฒ=๏„ž๐‘ฅ+12๐‘ฅโˆ’2๏€ผ12๐‘ฅ+2โˆ’12๐‘ฅโˆ’2๏ˆ๏Šช๏Šช
  • D๐‘ฆโ€ฒ=๏„ž๐‘ฅ+12๐‘ฅโˆ’2๏€พ12๐‘ฅ+2+4๐‘ฅ2๐‘ฅโˆ’2๏Š๏Šช๏Šฉ๏Šช
  • E๐‘ฆโ€ฒ=๏„ž๐‘ฅ+12๐‘ฅโˆ’2๏€พ12๐‘ฅ+2โˆ’8๐‘ฅ2๐‘ฅโˆ’2๏Š๏Šช๏Šฉ๏Šช

Q2:

Use logarithmic differentiation to find the derivative of the function ๐‘ฆ=2(๐‘ฅ)cos๏—.

  • A๐‘ฆโ€ฒ=[๐‘ฅ+๐‘ฅ๐‘ฅ][๐‘ฅโˆ’๐‘ฅ๐‘ฅ]lncoscotlncostan
  • B๐‘ฆโ€ฒ=๐‘ฅโˆ’๐‘ฅ๐‘ฅlncostan
  • C๐‘ฆโ€ฒ=2(๐‘ฅ)[๐‘ฅโˆ’๐‘ฅ๐‘ฅ]coslncostan๏—
  • D๐‘ฆโ€ฒ=2(๐‘ฅ)[๐‘ฅโˆ’๐‘ฅ๐‘ฅ]coslncoscot๏—
  • E๐‘ฆโ€ฒ=2(๐‘ฅ)[๐‘ฅ+๐‘ฅ๐‘ฅ]coslncostan๏—

Q3:

Use logarithmic differentiation to find the derivative of the function ๐‘ฆ=โˆ’5๐‘ฅ๏Šซ๏—cos.

  • A๐‘ฆโ€ฒ=๐‘ฅ๐‘ฅโˆ’๐‘ฅ๐‘ฅcossinln
  • B๐‘ฆโ€ฒ=25๐‘ฅ๏“๐‘ฅ๐‘ฅ+๐‘ฅ๐‘ฅ๏Ÿ๏Šซ๏—coscossinln
  • C๐‘ฆโ€ฒ=โˆ’25๐‘ฅ๏“๐‘ฅ๐‘ฅโˆ’๐‘ฅ๐‘ฅ๏Ÿ๏Šซ๏—coscossinln
  • D๐‘ฆโ€ฒ=25๐‘ฅ๏“๐‘ฅ๐‘ฅโˆ’๐‘ฅ๐‘ฅ๏Ÿ๏Šซ๏—coscossinln
  • E๐‘ฆโ€ฒ=โˆ’25๐‘ฅ๏“๐‘ฅ๐‘ฅ+๐‘ฅ๐‘ฅ๏Ÿ๏Šซ๏—coscossinln

Q4:

Use logarithmic differentiation to find the derivative of the function ๐‘ฆ=3(๐‘ฅ)tan๏Žข๏Žฃ๏‘.

  • A๐‘ฆโ€ฒ=9(๐‘ฅ)4๐‘ฅ๏–๐‘ฅ๐‘ฅโˆ’(๐‘ฅ)๐‘ฅ๏ขtansectanlntan๏Žข๏Žฃ๏‘๏Šจ
  • B๐‘ฆโ€ฒ=9(๐‘ฅ)4๐‘ฅ๏–๐‘ฅ๐‘ฅ+(๐‘ฅ)๐‘ฅ๏ขtansectanlntan๏Žข๏Žฃ๏‘๏Šจ
  • C๐‘ฆโ€ฒ=9(๐‘ฅ)4๐‘ฅ๏•๐‘ฅโˆ’(๐‘ฅ)๐‘ฅ๏กtanseclntan๏Žข๏Žฃ๏‘๏Šจ
  • D๐‘ฆโ€ฒ=34๐‘ฅ๏–๐‘ฅ๐‘ฅโˆ’(๐‘ฅ)๐‘ฅ๏ขsectanlntan๏Šจ
  • E๐‘ฆโ€ฒ=โˆ’9(๐‘ฅ)4๐‘ฅ๏–๐‘ฅ๐‘ฅ+(๐‘ฅ)๐‘ฅ๏ขtansectanlntan๏Žข๏Žฃ๏‘๏Šจ

Q5:

Use logarithmic differentiation to find the derivative of the function ๐‘ฆ=(๐‘ฅ)ln๏Šฉ๏—cos.

  • A๐‘ฆโ€ฒ=3(๐‘ฅ)๏“๐‘ฅ๐‘ฅ๐‘ฅโˆ’๐‘ฅ(๐‘ฅ)๏Ÿlncoslnsinlnln๏Šฉ๏—cos
  • B๐‘ฆโ€ฒ=(๐‘ฅ)3[๐‘ฅ๐‘ฅ๐‘ฅโˆ’๐‘ฅ(๐‘ฅ)]lnlncossinlnln๏Šฉ๏—cos
  • C๐‘ฆโ€ฒ=3(๐‘ฅ)๏“๐‘ฅ๐‘ฅ๐‘ฅ+๐‘ฅ(๐‘ฅ)๏Ÿlncoslnsinlnln๏Šฉ๏—cos
  • D๐‘ฆโ€ฒ=3(๐‘ฅ)[๐‘ฅ๐‘ฅ๐‘ฅโˆ’๐‘ฅ(๐‘ฅ)]lnlncossinlnln๏Šฉ๏—cos
  • E๐‘ฆโ€ฒ=3๏“๐‘ฅ๐‘ฅ๐‘ฅโˆ’๐‘ฅ(๐‘ฅ)๏Ÿcoslnsinlnln

Q6:

If โˆ’5๐‘ฆ=3๐‘ฅ๏Šฌ๏—, determine dd๐‘ฆ๐‘ฅ.

  • Aโˆ’185๐‘ฅ๏Šฌ๏—
  • Bโˆ’185๐‘ฅ(๐‘ฅ+1)๏Šฌ๏—ln
  • C18๐‘ฅ(3๐‘ฅ)ln
  • D35๐‘ฅ๏Šฌ๏—

Q7:

Determine dd๐‘ฆ๐‘ฅ, given that ๐‘ฆ=๏€น5๐‘ฅ+11๏…๏Šช๏Šฉ๏—.

  • A15๐‘ฅ๏€น5๐‘ฅ+11๏…(5๐‘ฅ+11)๏Šซ๏Šช๏Šช๏Šฉ๏—ln
  • B๏€น5๐‘ฅ+11๏…๏–3๏€น5๐‘ฅ+11๏…+60๐‘ฅ5๐‘ฅ+11๏ข๏Šช๏Šฉ๏—๏Šช๏Šช๏Šชln
  • Cln๏€น5๐‘ฅ+11๏…๐‘ฅ(5๐‘ฅ+11)๏Šช๏Šช๏Šฉ
  • D๏–3๏€น5๐‘ฅ+11๏…+15๐‘ฅ5๐‘ฅ+11๏ขln๏Šช๏Šช๏Šช
  • E๏€น5๐‘ฅ+11๏…๏–๏€น5๐‘ฅ๏…+15๐‘ฅ5๐‘ฅ+11๏ข๏Šช๏Šฉ๏—๏Šฉ๏Šช๏Šชln

Q8:

Find dd๐‘ฆ๐‘ฅ if ๐‘ฆ=๏€น6๐‘ฅ+7๏…๏Šฏ๏Šฎ๏—.

  • A8๏–๏€น6๐‘ฅ+7๏…+54๐‘ฅ6๐‘ฅ+7๏ขln๏Šฏ๏Šฏ๏Šฏ
  • B8๐‘ฆ๏–๏€น6๐‘ฅ+7๏…+54๐‘ฅ6๐‘ฅ+7๏ขln๏Šฏ๏Šง๏Šฆ๏Šฏ
  • C8๐‘ฆ๏–๏€น6๐‘ฅ+7๏…+54๐‘ฅ6๐‘ฅ+7๏ขln๏Šฏ๏Šฏ๏Šฏ
  • D8๐‘ฆ๏‘๏€น6๐‘ฅ+7๏…+54๐‘ฅ๏ln๏Šฏ๏Šฏ

Q9:

Determine dd๐‘ฆ๐‘ฅ, given that ๐‘ฆ=(84๐‘ฅ)sin๏Šจ๏—.

  • A16๐‘ฅ4๐‘ฅcos
  • B2(84๐‘ฅ)[(84๐‘ฅ)+4๐‘ฅ4๐‘ฅ]sinlnsincot๏Šจ๏—
  • C16๐‘ฅ(4๐‘ฅ)lncos
  • D(84๐‘ฅ)[(4๐‘ฅ)+๐‘ฅ4๐‘ฅ]sinlnsintan๏Šจ๏—

Q10:

Determine dd๐‘ฆ๐‘ฅ, if ๐‘ฆ=(54๐‘ฅ)sintan๏Šช๏—.

  • Atanlncos4๐‘ฅ(204๐‘ฅ)
  • B1+4๐‘ฅ(54๐‘ฅ)seclnsin๏Šจ
  • C4(54๐‘ฅ)๏‘1+4๐‘ฅ(54๐‘ฅ)๏sinseclnsintan๏Šช๏—๏Šจ
  • D204๐‘ฅ4๐‘ฅ+204๐‘ฅ4๐‘ฅsinseccostan๏Šจ

Q11:

Determine dd๐‘ฆ๐‘ฅ for the function ๐‘ฆ๐‘ฆ=(3๐‘ฅ+8):๏Šฑ๏Šจ๏Šฎ๏—cos.

  • A[3๐‘ฅ+8]๏”16(3๐‘ฅ+8)8๐‘ฅ+68๐‘ฅ3๐‘ฅ+8๏ ๏Šฑ๏Šจ๏Šฎ๏—coslnsincos
  • B[3๐‘ฅ+8]๏”โˆ’16(3๐‘ฅ+8)8๐‘ฅ+68๐‘ฅ3๐‘ฅ+8๏ ๏Šฑ๏Šจ๏Šฎ๏—coslnsincos
  • C[3๐‘ฅ+8]๏”16(3๐‘ฅ+8)8๐‘ฅโˆ’68๐‘ฅ3๐‘ฅ+8๏ ๏Šฑ๏Šจ๏Šฎ๏—coslnsincos
  • D16(3๐‘ฅ+8)8๐‘ฅโˆ’68๐‘ฅ3๐‘ฅ+8lnsincos

Q12:

Find dd๐‘ฆ๐‘ฅ, given that 7๐‘ฆ=6๐‘ฅsin๏Šฌ๏—.

  • A367๐‘ฅ๏€ฝ6๐‘ฅ๐‘ฅ+6๐‘ฅ6๐‘ฅ๏‰sin๏Šฌ๏—sinlncos
  • B67๐‘ฅ๏€ฝ6๐‘ฅ๐‘ฅ+6๐‘ฅ6๐‘ฅ๏‰sin๏Šฌ๏—sinlncos
  • Csincosln6๐‘ฅ๐‘ฅ+66๐‘ฅ๐‘ฅ
  • D67๐‘ฅ๏€ฝ6๐‘ฅ๐‘ฅโˆ’6๐‘ฅ6๐‘ฅ๏‰sin๏Šฌ๏—sinlncos
  • E67๐‘ฅ(๐‘ฅ6๐‘ฅ+66๐‘ฅ๐‘ฅ)sin๏Šฌ๏—sincosln

Q13:

Find dd๐‘ฆ๐‘ฅ if 6๐‘ฆ=7๐‘ฅ๏Žข๏Žค๏‘.

  • A710๐‘ฅ(1โˆ’๐‘ฅ)๏Žข๏Žค๏‘๏Šฑ๏Šจln
  • B710๐‘ฅ๏Žข๏Žค๏‘๏Šฑ๏Šจ
  • C710๐‘ฅ(1โˆ’๐‘ฅ)๏Šฑ๏Šจln
  • D35๐‘ฅ(1โˆ’๐‘ฅ)๏Žข๏Žค๏‘๏Šฑ๏Šจln
  • E710๐‘ฅ(1โˆ’๐‘ฅ)๏Žข๏Žค๏‘ln

Q14:

Given that ๐‘ฆ=(8๐‘ฅ)log๏Šช๏Šซ๏—tan, find dd๐‘ฆ๐‘ฅ.

  • A(8๐‘ฅ)๏”20(8๐‘ฅ)5๐‘ฅ+45๐‘ฅ๐‘ฅ10๏ loglnlogsectanln๏Šช๏Šซ๏—๏Šจtan
  • B20(8๐‘ฅ)5๐‘ฅ+45๐‘ฅ๐‘ฅ8๐‘ฅlnlogsectanlog๏Šจ
  • C20(8๐‘ฅ)5๐‘ฅ+45๐‘ฅ๐‘ฅ108๐‘ฅlnlogsectanlnlog๏Šจ
  • D(8๐‘ฅ)๏•20(8๐‘ฅ)5๐‘ฅ+45๐‘ฅ๐‘ฅ108๐‘ฅ๏กloglnlogsectanlnlog๏Šช๏Šซ๏—๏Šจtan

Q15:

Given that ๐‘ฆ=2๏Šฑ๏Šฏ๏Œพ๏Šฐ๏—๏Žจ๏‘sin, determine dd๐‘ฆ๐‘ฅ.

  • A๏€นโˆ’9๐‘’+๐‘ฅ๏…2๏Šฏ๏—cosln
  • B2๏€น81๐‘’โˆ’๐‘ฅ๏…2๏Šฑ๏Šฏ๏Œพ๏Šฐ๏—๏Šฏ๏—๏Žจ๏‘sincosln
  • C2๏€นโˆ’81๐‘’+๐‘ฅ๏…๏Šฑ๏Šฏ๏Œพ๏Šฐ๏—๏Šฏ๏—๏Žจ๏‘sincos
  • D2๏€นโˆ’81๐‘’+๐‘ฅ๏…2๏Šฑ๏Šฏ๏Œพ๏Šฐ๏—๏Šฏ๏—๏Žจ๏‘sincosln

Q16:

Given that ๐‘ฆ=(35๐‘ฅ)loglog๏Šซ๏—, find dd๐‘ฆ๐‘ฅ.

  • A๏€ฝ1+(35๐‘ฅ)๐‘ฅ10๏‰lnlogln
  • B๐‘ฆ๐‘ฅ10((35๐‘ฅ)+1)lnlnlog
  • C๐‘ฆ๏€ฝ1+(35๐‘ฅ)๐‘ฅ10๏‰lnlogln
  • D1๐‘ฅ10((35๐‘ฅ)+1)lnlnlog

Q17:

Given ๐‘ฆ=๐‘ฅ๏—๏‘, find dd๐‘ฆ๐‘ฅ.

  • A๐‘ฆ๐‘ฆ๏€ผ๐‘ฅ+1๐‘ฅ๐‘ฅ+1๏ˆlnlnln
  • Blnlnln๐‘ฆ๏€ผ๐‘ฅ+1๐‘ฅ๐‘ฅ+1๏ˆ
  • C๐‘ฆ๐‘ฆ๏€ผ1๐‘ฅ๐‘ฅ+1๏ˆlnln
  • D๐‘ฆ๐‘ฆ๏€ผ๐‘ฅ+1๐‘ฅ+1๏ˆlnlnln

Q18:

If ๐‘ฆ=๐‘’๏„ž๐‘ฅ+4โˆ’๐‘ฅ+4๏Šฑ๏Šฌ๏—, determine ๏€น16โˆ’๐‘ฅ๏…๐‘ฆโ€ฒ๏Šจ.

  • A๐‘ฆ๏€น6๐‘ฅโˆ’20๏…๏Šจ
  • B๐‘ฆ๏€น6๐‘ฅโˆ’92๏…๏Šจ
  • C6๐‘ฅ๐‘ฆ๏Šจ
  • D6๐‘ฅ+4๐‘ฆ๏Šจ
  • E๐‘ฆ๏€นโˆ’6๐‘ฅโˆ’92๏…๏Šจ

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