Worksheet: The Chain Rule

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

Q1:

Find the first derivative of the function 𝑦=5𝑥6.

  • A60𝑥5𝑥6
  • B65𝑥6
  • C60𝑥5𝑥6
  • D65𝑥6

Q2:

If 𝑦=8𝑥cos, find dd𝑦𝑥.

  • A40𝑥8𝑥cos
  • B40𝑥8𝑥cos
  • C8𝑥8𝑥cos
  • D40𝑥8𝑥sin
  • E8𝑥8𝑥sin

Q3:

Determine the derivative of 𝑦=2𝑥3𝑥+4.

  • A𝑦=4𝑥3𝑥2𝑥3𝑥+4
  • B𝑦=(4𝑥3)2𝑥3𝑥+4
  • C𝑦=55(4𝑥3)2𝑥3𝑥+4
  • D𝑦=55(4𝑥3)2𝑥3𝑥+4
  • E𝑦=554𝑥3𝑥2𝑥3𝑥+4

Q4:

Find the first derivative of the function 𝑦=8𝑥9𝑥sin.

  • A449𝑥9𝑥8𝑥9𝑥sincossin
  • B4369𝑥8𝑥9𝑥sinsin
  • C4+369𝑥9𝑥8𝑥9𝑥sincossin
  • D4369𝑥9𝑥8𝑥9𝑥sincossin

Q5:

If 𝑦=8𝑥5𝑥sin, find dd𝑦𝑥.

  • A8𝑥5𝑥205𝑥5𝑥+8sinsincos
  • B205𝑥5𝑥+8sincos
  • C105𝑥5𝑥+48𝑥5𝑥sincossin
  • D105𝑥5𝑥+48𝑥5𝑥sincossin
  • E205𝑥5𝑥+88𝑥5𝑥sincossin

Q6:

If 𝑦=𝑥9𝑥+5cos, find dd𝑦𝑥.

  • A7𝑥9𝑥+5𝑥9𝑥+5sincos
  • B35(9𝑥+5)sincos
  • C7𝑥9𝑥+5cos
  • D5(9𝑥+5)sincos
  • E35(9𝑥+5)sincos

Q7:

If 𝑦=8(6𝑥)(6𝑥)sinsincossin, find dd𝑦𝑥.

  • A66𝑥8(6𝑥)+(6𝑥)coscossinsinsin
  • B66𝑥8(6𝑥)+(6𝑥)coscossinsinsin
  • Ccoscossincossin6𝑥8(6𝑥)+(6𝑥)
  • D66𝑥8(6𝑥)(6𝑥)coscossincossin

Q8:

Find the derivative of the function 𝑦=(𝜋𝑥)cossintan.

  • A𝑦=𝜋(𝜋𝑥)(𝜋𝑥)2(𝜋𝑥)costansinsintansintan
  • B𝑦=2𝜋(𝜋𝑥)(𝜋𝑥)(𝜋𝑥)(𝜋𝑥)costansecsinsintansintan
  • C𝑦=𝜋(𝜋𝑥)(𝜋𝑥)(𝜋𝑥)2(𝜋𝑥)costansecsinsintansintan
  • D𝑦=𝜋(𝜋𝑥)(𝜋𝑥)(𝜋𝑥)2(𝜋𝑥)costansecsinsintansintan
  • E𝑦=𝜋(𝜋𝑥)(𝜋𝑥)(𝜋𝑥)(𝜋𝑥)costansecsinsintansintan

Q9:

If 𝑦=𝑥sincos, find dd𝑦𝑥.

  • A3𝑥𝑥𝑥sincoscoscos
  • B3𝑥𝑥sincos
  • C3𝑥𝑥𝑥sincoscoscos
  • D3𝑥𝑥sincos
  • E3𝑥𝑥coscoscos

Q10:

If 𝑦=(8𝑥4)sin, find dd𝑦𝑥.

  • A(8𝑥4)cos
  • B16𝑥(8𝑥4)cos
  • C16𝑥(8𝑥4)sin
  • D16𝑥(8𝑥4)cos

Q11:

If 𝑦=66𝑥11tan, find dd𝑦𝑥.

  • A72𝑥6𝑥11sec
  • B72𝑥6𝑥11sec
  • C72𝑥6𝑥11sec
  • D726𝑥11sec
  • E66𝑥11sec

Q12:

If 𝑦=52𝑥cos, find dd𝑦𝑥.

  • A302𝑥2𝑥cossin
  • B302𝑥2𝑥cossin
  • C302𝑥sin
  • D52𝑥2𝑥cossin
  • E302𝑥cos

Q13:

If 𝑦=(125𝑥)coscos, find dd𝑦𝑥.

  • A12(125𝑥)5𝑥sincossin
  • B(125𝑥)sincos
  • C60(125𝑥)5𝑥sincossin
  • D12(125𝑥)5𝑥sincossin
  • E60(125𝑥)5𝑥sincossin

Q14:

If 𝑦=2+4𝑥tan, find dd𝑦𝑥.

  • A22432+4𝑥𝑥𝑥tantansec
  • B22432+4𝑥𝑥𝑥tantansec
  • C22432+4𝑥𝑥𝑥tantansec
  • D732+4𝑥𝑥𝑥tantansec

Q15:

If 𝑦=(27𝑥+27𝑥)sincos, find dd𝑦𝑥.

  • A56(7𝑥+7𝑥)sincos
  • B47𝑥+47𝑥sincos
  • C814𝑥cos
  • D5614𝑥cos

Q16:

If 𝑦=17𝑥sin, find dd𝑦𝑥.

  • A17𝑥17𝑥cos
  • Bcos17𝑥
  • C17𝑥17𝑥sin
  • D153𝑥17𝑥cos

Q17:

If 𝑦=9𝑥tan, find dd𝑦𝑥.

  • A9𝑥2𝑥sec
  • B9𝑥sec
  • C9𝑥2𝑥sec
  • D9𝑥2𝑥sec

Q18:

If 𝑦=810𝑥cos, find dd𝑦𝑥.

  • A810𝑥10𝑥sincos
  • B8010𝑥10𝑥sincos
  • C4010𝑥10𝑥sincos
  • D4010𝑥cos
  • E8010𝑥10𝑥sincos

Q19:

If 𝑦=9𝑥tan, find dd𝑦𝑥.

  • A819𝑥9𝑥tansec
  • B819𝑥9𝑥tansec
  • C819𝑥9𝑥tansec
  • D819𝑥9𝑥tansec
  • E99𝑥9𝑥tansec

Q20:

If 𝑦=6𝑥7+9𝑥tan, find dd𝑦𝑥.

  • A86𝑥7+9𝑥6𝑥7+9𝑥tansec
  • B242+9𝑥6𝑥7+9𝑥sec
  • C242+9𝑥6𝑥7+9𝑥6𝑥7+9𝑥tansec
  • D242+9𝑥6𝑥7+9𝑥tan
  • E242+9𝑥6𝑥7+9𝑥6𝑥7+9𝑥tansec

Q21:

If 𝑦=5(4𝑥)sintan, find dd𝑦𝑥.

  • A5(4𝑥)4𝑥costansec
  • B20(4𝑥)4𝑥costansec
  • C20(4𝑥)costan
  • D20(4𝑥)4𝑥costansec
  • E20(4𝑥)4𝑥costansec

Q22:

Find the derivative of the function 𝑦=𝑥+𝑥+𝑥.

  • A𝑦=2𝑥+𝑥+12𝑥+𝑥+𝑥𝑥+𝑥
  • B𝑦=4𝑥𝑥+𝑥+2𝑥+14𝑥𝑥+𝑥𝑥+𝑥+𝑥
  • C𝑦=2𝑥+𝑥+14𝑥+𝑥+𝑥𝑥+𝑥
  • D𝑦=12𝑥+𝑥+𝑥
  • E𝑦=4𝑥𝑥+𝑥+2𝑥+18𝑥𝑥+𝑥𝑥+𝑥+𝑥

Q23:

Find dd𝑦𝑥, given that 𝑦=(13𝑥).

  • A1172(13𝑥)
  • B92(13𝑥)
  • C1172(13𝑥)
  • D1172(13𝑥)

Q24:

Given that 𝑦=𝑓(𝑥), 𝑓(4)=2, and 𝑓(4)=7, determine dd𝑦𝑥 at 𝑥=4.

  • A17
  • B277
  • C27
  • D77

Q25:

Find dd𝑥2𝑥+2𝑥 at 𝑥=1.

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