Lesson Worksheet: The Chain Rule Mathematics • Higher Education

In this worksheet, we will practice finding the derivatives of composite functions using the chain rule.

Q1:

Find the first derivative of the function ๐‘ฆ=๏€น5๐‘ฅโˆ’6๏…๏Šจ๏Šฌ.

  • A60๐‘ฅ๏€น5๐‘ฅโˆ’6๏…๏Šจ๏Šซ
  • B6๏€น5๐‘ฅโˆ’6๏…๏Šจ๏Šซ
  • C60๐‘ฅ๏€น5๐‘ฅโˆ’6๏…๏Šจ๏Šญ
  • D6๏€น5๐‘ฅโˆ’6๏…๏Šจ๏Šญ

Q2:

If ๐‘ฆ=๏€ผ8๐‘ฅ๏ˆcos๏Šซ, find dd๐‘ฆ๐‘ฅ.

  • Aโˆ’40๐‘ฅ๏€ผ8๐‘ฅ๏ˆ๏Šซ๏Šซcos
  • Bโˆ’40๐‘ฅ๏€ผ8๐‘ฅ๏ˆ๏Šฌ๏Šซcos
  • Cโˆ’8๐‘ฅ๏€ผ8๐‘ฅ๏ˆ๏Šซ๏Šซcos
  • D40๐‘ฅ๏€ผ8๐‘ฅ๏ˆ๏Šฌ๏Šซsin
  • Eโˆ’8๐‘ฅ๏€ผ8๐‘ฅ๏ˆ๏Šซ๏Šซsin

Q3:

Determine the derivative of ๐‘ฆ=๏€นโˆ’2๐‘ฅโˆ’3๐‘ฅ+4๏…๏Šจ๏Šซ๏Šซ.

  • A๐‘ฆโ€ฒ=๏€นโˆ’4๐‘ฅโˆ’3๐‘ฅ๏…๏€นโˆ’2๐‘ฅโˆ’3๐‘ฅ+4๏…๏Šจ๏Šจ๏Šซ๏Šซ
  • B๐‘ฆโ€ฒ=(โˆ’4๐‘ฅโˆ’3)๏€นโˆ’2๐‘ฅโˆ’3๐‘ฅ+4๏…๏Šจ๏Šซ๏Šช
  • C๐‘ฆโ€ฒ=55(โˆ’4๐‘ฅโˆ’3)๏€นโˆ’2๐‘ฅโˆ’3๐‘ฅ+4๏…๏Šจ๏Šซ๏Šซ
  • D๐‘ฆโ€ฒ=55(โˆ’4๐‘ฅโˆ’3)๏€นโˆ’2๐‘ฅโˆ’3๐‘ฅ+4๏…๏Šจ๏Šซ๏Šช
  • E๐‘ฆโ€ฒ=55๏€นโˆ’4๐‘ฅโˆ’3๐‘ฅ๏…๏€นโˆ’2๐‘ฅโˆ’3๐‘ฅ+4๏…๏Šจ๏Šจ๏Šซ๏Šช

Q4:

Find the first derivative of the function ๐‘ฆ=โˆš8๐‘ฅโˆ’9๐‘ฅsin๏Šฎ.

  • A4โˆ’49๐‘ฅ9๐‘ฅโˆš8๐‘ฅโˆ’9๐‘ฅsincossin๏Šญ๏Šฎ
  • B4โˆ’369๐‘ฅโˆš8๐‘ฅโˆ’9๐‘ฅsinsin๏Šญ๏Šฎ
  • C4+369๐‘ฅ9๐‘ฅโˆš8๐‘ฅโˆ’9๐‘ฅsincossin๏Šญ๏Šฎ
  • D4โˆ’369๐‘ฅ9๐‘ฅโˆš8๐‘ฅโˆ’9๐‘ฅsincossin๏Šญ๏Šฎ

Q5:

If ๐‘ฆ=โˆš8๐‘ฅโˆ’5๐‘ฅsin๏Šช, find dd๐‘ฆ๐‘ฅ.

  • Aโˆš8๐‘ฅโˆ’5๐‘ฅ๏€บโˆ’205๐‘ฅ5๐‘ฅ+8๏†sinsincos๏Šช๏Šฉ
  • Bโˆ’205๐‘ฅ5๐‘ฅ+8sincos๏Šฉ
  • C105๐‘ฅ5๐‘ฅ+4โˆš8๐‘ฅ5๐‘ฅsincossin๏Šฉ๏Šช
  • Dโˆ’105๐‘ฅ5๐‘ฅ+4โˆš8๐‘ฅโˆ’5๐‘ฅsincossin๏Šฉ๏Šช
  • Eโˆ’205๐‘ฅ5๐‘ฅ+8โˆš8๐‘ฅโˆ’5๐‘ฅsincossin๏Šฉ๏Šช

Q6:

If ๐‘ฆ=โˆ’8(6๐‘ฅ)โˆ’(6๐‘ฅ)sinsincossin, find dd๐‘ฆ๐‘ฅ.

  • Aโˆ’66๐‘ฅ๏‘โˆ’8(6๐‘ฅ)+(6๐‘ฅ)๏coscossinsinsin
  • B66๐‘ฅ๏‘โˆ’8(6๐‘ฅ)+(6๐‘ฅ)๏coscossinsinsin
  • Ccoscossincossin6๐‘ฅ๏‘โˆ’8(6๐‘ฅ)+(6๐‘ฅ)๏
  • Dโˆ’66๐‘ฅ๏‘โˆ’8(6๐‘ฅ)โˆ’(6๐‘ฅ)๏coscossincossin

Q7:

Find the derivative of the function ๐‘ฆ=โˆš(๐œ‹๐‘ฅ)cossintan.

  • A๐‘ฆโ€ฒ=๐œ‹(๐œ‹๐‘ฅ)โˆš(๐œ‹๐‘ฅ)2โˆš(๐œ‹๐‘ฅ)costansinsintansintan
  • B๐‘ฆโ€ฒ=2๐œ‹(๐œ‹๐‘ฅ)(๐œ‹๐‘ฅ)โˆš(๐œ‹๐‘ฅ)โˆš(๐œ‹๐‘ฅ)costansecsinsintansintan๏Šจ
  • C๐‘ฆโ€ฒ=โˆ’๐œ‹(๐œ‹๐‘ฅ)(๐œ‹๐‘ฅ)โˆš(๐œ‹๐‘ฅ)2โˆš(๐œ‹๐‘ฅ)costansecsinsintansintan๏Šจ
  • D๐‘ฆโ€ฒ=๐œ‹(๐œ‹๐‘ฅ)(๐œ‹๐‘ฅ)โˆš(๐œ‹๐‘ฅ)2โˆš(๐œ‹๐‘ฅ)costansecsinsintansintan๏Šจ
  • E๐‘ฆโ€ฒ=๐œ‹(๐œ‹๐‘ฅ)(๐œ‹๐‘ฅ)โˆš(๐œ‹๐‘ฅ)โˆš(๐œ‹๐‘ฅ)costansecsinsintansintan๏Šจ

Q8:

If ๐‘ฆ=๏€น๐‘ฅ๏…sincos๏Šฉ, find dd๐‘ฆ๐‘ฅ.

  • Aโˆ’3๐‘ฅ๐‘ฅ๏€น๐‘ฅ๏…sincoscoscos๏Šจ๏Šฉ
  • Bโˆ’3๐‘ฅ๐‘ฅsincos๏Šจ
  • C3๐‘ฅ๐‘ฅ๏€น๐‘ฅ๏…sincoscoscos๏Šจ๏Šฉ
  • Dโˆ’3๐‘ฅ๐‘ฅsincos๏Šฉ
  • Eโˆ’3๐‘ฅ๏€น๐‘ฅ๏…coscoscos๏Šจ๏Šฉ

Q9:

If ๐‘ฆ=(8๐‘ฅโˆ’4)sin๏Šจ, find dd๐‘ฆ๐‘ฅ.

  • Aโˆ’(8๐‘ฅโˆ’4)cos๏Šจ
  • B16๐‘ฅ(8๐‘ฅโˆ’4)cos๏Šจ
  • C16๐‘ฅ(8๐‘ฅโˆ’4)sin๏Šจ
  • Dโˆ’16๐‘ฅ(8๐‘ฅโˆ’4)cos๏Šจ

Q10:

If ๐‘ฆ=6๏€น6๐‘ฅโˆ’11๏…tan๏Šจ, find dd๐‘ฆ๐‘ฅ.

  • A72๐‘ฅ๏€น6๐‘ฅโˆ’11๏…sec๏Šจ๏Šจ
  • B72๐‘ฅ๏€น6๐‘ฅโˆ’11๏…sec๏Šจ
  • Cโˆ’72๐‘ฅ๏€น6๐‘ฅโˆ’11๏…sec๏Šจ๏Šจ
  • D72๏€น6๐‘ฅโˆ’11๏…sec๏Šจ๏Šจ
  • E6๏€น6๐‘ฅโˆ’11๏…sec๏Šจ๏Šจ

Q11:

If ๐‘ฆ=52๐‘ฅcos๏Šฉ, find dd๐‘ฆ๐‘ฅ.

  • Aโˆ’302๐‘ฅ2๐‘ฅcossin๏Šจ
  • Bโˆ’302๐‘ฅ2๐‘ฅcossin๏Šช
  • C302๐‘ฅsin๏Šจ
  • Dโˆ’52๐‘ฅ2๐‘ฅcossin๏Šจ
  • E302๐‘ฅcos๏Šจ

Q12:

If ๐‘ฆ=(125๐‘ฅ)coscos, find dd๐‘ฆ๐‘ฅ.

  • Aโˆ’12(125๐‘ฅ)5๐‘ฅsincossin
  • Bโˆ’(125๐‘ฅ)sincos
  • Cโˆ’60(125๐‘ฅ)5๐‘ฅsincossin
  • D12(125๐‘ฅ)5๐‘ฅsincossin
  • E60(125๐‘ฅ)5๐‘ฅsincossin

Q13:

If ๐‘ฆ=๏„๏€น2+4๐‘ฅ๏…๏Žขtan๏Šญ๏Šฎ, find dd๐‘ฆ๐‘ฅ.

  • A2243๏„๏€น2+4๐‘ฅ๏…๐‘ฅ๐‘ฅ๏Žขtantansec๏Šญ๏Šฎ๏Šฎ
  • B2243๏„๏€น2+4๐‘ฅ๏…๐‘ฅ๐‘ฅ๏Žขtantansec๏Šญ๏Šซ๏Šฌ๏Šจ
  • C2243๏„๏€น2+4๐‘ฅ๏…๐‘ฅ๐‘ฅ๏Žขtantansec๏Šญ๏Šซ๏Šญ
  • D73๏„๏€น2+4๐‘ฅ๏…๐‘ฅ๐‘ฅ๏Žขtantansec๏Šญ๏Šง๏Šง๏Šฌ๏Šจ

Q14:

If ๐‘ฆ=(27๐‘ฅ+27๐‘ฅ)sincos๏Šจ, find dd๐‘ฆ๐‘ฅ.

  • A56(7๐‘ฅ+7๐‘ฅ)sincos๏Šจ
  • B47๐‘ฅ+47๐‘ฅsincos
  • C814๐‘ฅcos
  • D5614๐‘ฅcos

Q15:

If ๐‘ฆ=๏€ผ17๐‘ฅ๏ˆsin๏Šฏ, find dd๐‘ฆ๐‘ฅ.

  • A17๐‘ฅ๏€ผ17๐‘ฅ๏ˆ๏Šฏ๏Šฏcos
  • Bcos๏€ผ17๐‘ฅ๏ˆ๏Šฏ
  • Cโˆ’17๐‘ฅ๏€ผ17๐‘ฅ๏ˆ๏Šง๏Šฆ๏Šฏsin
  • Dโˆ’153๐‘ฅ๏€ผ17๐‘ฅ๏ˆ๏Šง๏Šฆ๏Šฏcos

Q16:

If ๐‘ฆ=9โˆš๐‘ฅtan, find dd๐‘ฆ๐‘ฅ.

  • Aโˆ’9โˆš๐‘ฅ2โˆš๐‘ฅsec๏Šจ
  • B9โˆš๐‘ฅsec๏Šจ
  • C9โˆš๐‘ฅ2โˆš๐‘ฅsec๏Šจ
  • D9โˆš๐‘ฅ2โˆš๐‘ฅsec

Q17:

If ๐‘ฆ=8โˆš10๐‘ฅcos, find dd๐‘ฆ๐‘ฅ.

  • Aโˆ’810๐‘ฅโˆš10๐‘ฅsincos
  • B8010๐‘ฅโˆš10๐‘ฅsincos
  • Cโˆ’4010๐‘ฅโˆš10๐‘ฅsincos
  • Dโˆ’40โˆš10๐‘ฅcos
  • Eโˆ’8010๐‘ฅโˆš10๐‘ฅsincos๏Šฉ

Q18:

If ๐‘ฆ=9๐‘ฅtan๏Šฏ, find dd๐‘ฆ๐‘ฅ.

  • A819๐‘ฅ9๐‘ฅtansec๏Šฎ๏Šจ
  • Bโˆ’819๐‘ฅ9๐‘ฅtansec๏Šฎ๏Šจ
  • C819๐‘ฅ9๐‘ฅtansec๏Šฎ
  • D819๐‘ฅ9๐‘ฅtansec๏Šฏ๏Šจ
  • E99๐‘ฅ9๐‘ฅtansec๏Šฎ๏Šจ

Q19:

If ๐‘ฆ=๏€น6๐‘ฅโˆ’7+9๐‘ฅ๏…tan๏Šฎ๏Šฉ, find dd๐‘ฆ๐‘ฅ.

  • A8๏€น6๐‘ฅโˆ’7+9๐‘ฅ๏…๏€น6๐‘ฅโˆ’7+9๐‘ฅ๏…tansec๏Šญ๏Šฉ๏Šจ๏Šฉ
  • B24๏€น2+9๐‘ฅ๏…๏€น6๐‘ฅโˆ’7+9๐‘ฅ๏…๏Šจ๏Šจ๏Šฉsec
  • C24๏€น2+9๐‘ฅ๏…๏€น6๐‘ฅโˆ’7+9๐‘ฅ๏…๏€น6๐‘ฅโˆ’7+9๐‘ฅ๏…๏Šจ๏Šญ๏Šฉ๏Šจ๏Šฉtansec
  • D24๏€น2+9๐‘ฅ๏…๏€น6๐‘ฅโˆ’7+9๐‘ฅ๏…๏Šจ๏Šญ๏Šฉtan
  • E24๏€น2+9๐‘ฅ๏…๏€น6๐‘ฅโˆ’7+9๐‘ฅ๏…๏€น6๐‘ฅโˆ’7+9๐‘ฅ๏…๏Šจ๏Šญ๏Šฉ๏Šฉtansec

Q20:

If ๐‘ฆ=5(4๐‘ฅ)sintan, find dd๐‘ฆ๐‘ฅ.

  • A5(4๐‘ฅ)4๐‘ฅcostansec๏Šจ
  • B20(4๐‘ฅ)4๐‘ฅcostansec
  • C20(4๐‘ฅ)costan
  • Dโˆ’20(4๐‘ฅ)4๐‘ฅcostansec๏Šจ
  • E20(4๐‘ฅ)4๐‘ฅcostansec๏Šจ

Q21:

Find the derivative of the function ๐‘ฆ=๏„ž๐‘ฅ+๏„๐‘ฅ+โˆš๐‘ฅ.

  • A๐‘ฆโ€ฒ=2๏„๐‘ฅ+โˆš๐‘ฅ+12๏„ž๐‘ฅ+๏„๐‘ฅ+โˆš๐‘ฅ๏„๐‘ฅ+โˆš๐‘ฅ
  • B๐‘ฆโ€ฒ=4โˆš๐‘ฅ๏„๐‘ฅ+โˆš๐‘ฅ+2โˆš๐‘ฅ+14โˆš๐‘ฅ๏„๐‘ฅ+โˆš๐‘ฅ๏„ž๐‘ฅ+๏„๐‘ฅ+โˆš๐‘ฅ
  • C๐‘ฆโ€ฒ=2๏„๐‘ฅ+โˆš๐‘ฅ+14๏„ž๐‘ฅ+๏„๐‘ฅ+โˆš๐‘ฅ๏„๐‘ฅ+โˆš๐‘ฅ
  • D๐‘ฆโ€ฒ=12๏„ž๐‘ฅ+๏„๐‘ฅ+โˆš๐‘ฅ
  • E๐‘ฆโ€ฒ=4โˆš๐‘ฅ๏„๐‘ฅ+โˆš๐‘ฅ+2โˆš๐‘ฅ+18โˆš๐‘ฅ๏„๐‘ฅ+โˆš๐‘ฅ๏„ž๐‘ฅ+๏„๐‘ฅ+โˆš๐‘ฅ

Q22:

Find dd๐‘ฆ๐‘ฅ, given that ๐‘ฆ=โˆš(13๐‘ฅ)๏Šฏ.

  • A1172โˆš(13๐‘ฅ)๏Šฏ
  • B92โˆš(13๐‘ฅ)๏Šญ
  • C1172โˆš(13๐‘ฅ)๏Šญ
  • D1172โˆš(13๐‘ฅ)๏Šง๏Šง

Q23:

Given that ๐‘ฆ=๏„๐‘“(๐‘ฅ), ๐‘“โ€ฒ(4)=2, and ๐‘“(4)=7, determine dd๐‘ฆ๐‘ฅ at ๐‘ฅ=4.

  • A17
  • B2โˆš77
  • C27
  • Dโˆš77

Q24:

Find dd๐‘ฅ๏€ฟ2โˆš๐‘ฅ+2โˆš๐‘ฅ๏‹๏Šจ at ๐‘ฅ=1.

Q25:

Evaluate ๐‘“โ€ฒ(โˆ’1), where ๐‘“(๐‘ฅ)=โˆšโˆ’8๐‘ฅ+13๏Šจ.

  • A16โˆš55
  • Bโˆ’16โˆš55
  • C8โˆš5
  • D8โˆš55
  • Eโˆš510

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