Worksheet: Combining the Product, Quotient, and Chain Rules

In this worksheet, we will practice finding the first derivative of a function using combinations of the product, quotient, and chain rules.

Q1:

Find the first derivative of the function đ‘Ļ=9đ‘Ĩ+5đ‘Ĩī€ŧ4đ‘Ĩ+5đ‘ĨīˆīŠ¨īŠ¨.

  • A400đ‘Ĩ+400đ‘Ĩ+125đ‘Ĩ+9īŠĒīŠąīŠ¨
  • B320đ‘Ĩ+200đ‘Ĩ−250đ‘Ĩ+9īŠĒīŠąīŠ¨
  • C80đ‘Ĩ+200đ‘Ĩ+125đ‘Ĩ+9īŠĒīŠąīŠ¨
  • D400đ‘Ĩ+400đ‘Ĩ−125đ‘Ĩ+9īŠĒīŠąīŠ¨

Q2:

Differentiate đ‘Ļ=√đ‘Ĩ(−2đ‘Ĩ+1)īŽĸ.

  • Ađ‘Ļ′=−2đ‘Ĩ+đ‘ĨīŽĻīŽĸīŽŖīŽĸ
  • Bđ‘Ļ′=−2√đ‘Ĩ3−23đ‘ĨīŽĸīŽĄīŽĸ
  • Cđ‘Ļ′=−2√đ‘Ĩ+1đ‘ĨīŽĸīŽĄīŽĸ
  • Dđ‘Ļ′=−8đ‘Ĩ3+đ‘Ĩ3īŽĻīŽĸīŽŖīŽĸ
  • Eđ‘Ļ′=−8√đ‘Ĩ3+13đ‘ĨīŽĸīŽĄīŽĸ

Q3:

Find the first derivative of đ‘Ļ=ī„ž9đ‘Ĩ−49đ‘Ĩ+4secsec.

  • A369đ‘Ĩ(9đ‘Ĩ−4)(9đ‘Ĩ+4)ī„ž9đ‘Ĩ−49đ‘Ĩ+4tansecsecsecsecīŠ¨
  • B369đ‘Ĩ9đ‘Ĩ(9đ‘Ĩ−4)(9đ‘Ĩ+4)ī„ž9đ‘Ĩ−49đ‘Ĩ+4sectansecsecsecsecīŠ¨īŠ¨
  • C−369đ‘Ĩ9đ‘Ĩ(9đ‘Ĩ−4)(9đ‘Ĩ+4)ī„ž9đ‘Ĩ−49đ‘Ĩ+4sectansecsecsecsec
  • D369đ‘Ĩ9đ‘Ĩ(9đ‘Ĩ−4)(9đ‘Ĩ+4)ī„ž9đ‘Ĩ−49đ‘Ĩ+4sectansecsecsecsec

Q4:

Determine the derivative of the function 𝑠(𝑡)=ī„žâˆ’đ‘Ą+7−𝑡+7sincos.

  • A𝑠′(𝑡)=−7𝑡+7𝑡+12√−𝑡+7(−𝑡+7)sincossincosīŽĄīŽĸ
  • B𝑠′(𝑡)=7𝑡+7𝑡−12√−𝑡+7(−𝑡+7)sincossincosīŽĸīŽĄ
  • C𝑠′(𝑡)=−7𝑡+7𝑡+12√−𝑡+7(−𝑡+7)sincossincosīŽĸīŽĄ
  • D𝑠′(𝑡)=−7𝑡+7𝑡−12√−𝑡+7(−𝑡+7)sincossincosīŽĸīŽĄ
  • E𝑠′(𝑡)=−7𝑡+7𝑡−12√−𝑡+7(−𝑡+7)sincossincosīŽĄīŽĸ

Q5:

Let 𝑔(đ‘Ĩ)=ℎ[6đ‘Ĩ−𝑓(đ‘Ĩ)]. If 𝑓(−4)=−5, 𝑓′(−4)=2, and ℎ′(−19)=−2, find 𝑔′(−4).

Q6:

Find ddđ‘Ļđ‘Ĩ at đ‘Ĩ=2 when đ‘Ļ=ī€šđ‘Ĩ+đ‘Ĩ−2ī…ī€šâˆ’3đ‘Ĩ+7đ‘Ĩ−1ī…īŠ¨īŠ¨īŠ¨īŠĢ.

Q7:

Find the first derivative of đ‘Ļ=(đ‘Ĩ−5)(đ‘Ĩ−2)īŠŦ at (1,−4).

Q8:

Find the first derivative of the function đ‘Ļ=đ‘Ĩ(4đ‘Ĩ+9)īŠĒīŠ¯ at đ‘Ĩ=−2.

Q9:

Determine the derivative of ℎ(𝑡)=(2𝑡−5)ī€šâˆ’4𝑡+4ī…īŽĸīŽ¤īŠ¨īŠŠ.

  • Aℎ′(𝑡)=−24𝑡(2𝑡−5)ī€šâˆ’4𝑡+4ī…+6ī€šâˆ’4𝑡+4ī…5(2𝑡−5)īŽĸīŽ¤īŽĄīŽ¤īŠ¨īŠ¨īŠ¨īŠŠ
  • Bℎ′(𝑡)=24𝑡(2𝑡−5)ī€šâˆ’4𝑡+4ī…+6ī€šâˆ’4𝑡+4ī…5(2𝑡−5)īŽĸīŽ¤īŽĄīŽ¤īŠ¨īŠ¨īŠ¨īŠŠ
  • Cℎ′(𝑡)=−8𝑡(2𝑡−5)ī€šâˆ’4𝑡+4ī…+2ī€šâˆ’4𝑡+4ī…(2𝑡−5)īŽĸīŽ¤īŽĄīŽ¤īŠ¨īŠ¨īŠ¨īŠŠ
  • Dℎ′(𝑡)=3(2𝑡−5)ī€šâˆ’4𝑡+4ī…−625ī€šâˆ’4𝑡+4ī…īŽĸīŽ¤īŠ¨īŠ¨īŠ¨īŠŠ
  • Eℎ′(𝑡)=−3(2𝑡−5)ī€šâˆ’4𝑡+4ī…−625ī€šâˆ’4𝑡+4ī…īŽĸīŽ¤īŠ¨īŠ¨īŠ¨īŠŠ

Q10:

Find ddđ‘Ĩī€ģ5đ‘Ĩ√2đ‘Ĩ+2ī‡īŠ¨ at đ‘Ĩ=1.

Q11:

Evaluate the first derivative of đ‘Ļ=4đ‘Ĩ√đ‘Ĩ+6 at (−2,−16).

  • A6
  • B8
  • C0
  • D10
  • E1

Q12:

Determine the derivative of 𝐹(𝑡)=(−3𝑡+1)(4𝑡−2)īŠĢīŠąīŠŠ.

  • A𝐹′(𝑡)=−3(−3𝑡+1)(4𝑡−2)+5(−3𝑡+1)(4𝑡−2)īŠĢīŠĒīŠĒīŠŠ
  • B𝐹′(𝑡)=12(−3𝑡+1)(4𝑡−2)−15(−3𝑡+1)(4𝑡−2)īŠĢīŠĒīŠĒīŠŠ
  • C𝐹′(𝑡)=−12(−3𝑡+1)(4𝑡−2)−15(−3𝑡+1)(4𝑡−2)īŠĢīŠĒīŠĒīŠŠ
  • D𝐹′(𝑡)=3(−3𝑡+1)(4𝑡−2)+5(−3𝑡+1)(4𝑡−2)īŠĢīŠĒīŠĒīŠŠ
  • E𝐹′(𝑡)=−12(−3𝑡+1)(4𝑡−2)−15(−3𝑡+1)(4𝑡−2)īŠĢīŠ¨īŠĒīŠŠ

Q13:

Determine the derivative of 𝑔(đ‘Ĩ)=(−đ‘Ĩ+1)ī€š3đ‘Ĩ−đ‘Ĩ−2ī…īŠŠīŠ¨īŠŦ.

  • A𝑔′(đ‘Ĩ)=(−đ‘Ĩ+1)ī€šâˆ’45đ‘Ĩ+27đ‘Ĩī…ī€š3đ‘Ĩ−đ‘Ĩ−2ī…īŠ¨īŠ¨īŠ¨īŠĢ
  • B𝑔′(đ‘Ĩ)=(−đ‘Ĩ+1)ī€šâˆ’45đ‘Ĩ+45đ‘Ĩī…ī€š3đ‘Ĩ−đ‘Ĩ−2ī…īŠ¨īŠ¨īŠ¨īŠĢ
  • C𝑔′(đ‘Ĩ)=(−đ‘Ĩ+1)ī€šâˆ’45đ‘Ĩ−45đ‘Ĩī…ī€š3đ‘Ĩ−đ‘Ĩ−2ī…īŠ¨īŠ¨īŠ¨īŠĢ
  • D𝑔′(đ‘Ĩ)=(−đ‘Ĩ+1)ī€šâˆ’45đ‘Ĩ+45đ‘Ĩī…ī€š3đ‘Ĩ−đ‘Ĩ−2ī…īŠ¨īŠ¨īŠ¨īŠŦ
  • E𝑔′(đ‘Ĩ)=−3(−đ‘Ĩ+1)(6đ‘Ĩ−1)ī€š3đ‘Ĩ−đ‘Ĩ−2ī…īŠ¨īŠ¨īŠĢ

Q14:

Find the first derivative of the function đ‘Ļ=ī€š9đ‘Ĩ−7ī…√2đ‘Ĩ+1īŠ¨ at đ‘Ĩ=0.

  • A7
  • B−632
  • C−7
  • D9

Q15:

If đ‘Ļ=ī„ž2đ‘Ĩ+12đ‘Ĩ−1īŠŠīŠŠ, determine ddđ‘Ļđ‘Ĩ.

  • A−6đ‘Ĩ4đ‘Ĩ+1īŠ¨īŠŦ
  • B−6đ‘Ĩ4đ‘Ĩ−1īŠ¨īŠ¯
  • C−6đ‘Ĩ4đ‘Ĩ−1īŠ¨īŠŦ
  • D−12đ‘Ĩ4đ‘Ĩ−1īŠ¨īŠŦ

Q16:

Find the derivative of the function đ‘Ļ=ī„žđ‘Ĩ−5đ‘Ĩ−2.

  • Ađ‘Ļ′=−1√đ‘Ĩ(−5đ‘Ĩ−2)īŽĸīŽĄ
  • Bđ‘Ļ′=−1√đ‘Ĩ(−5đ‘Ĩ−2)īŽĄīŽĸ
  • Cđ‘Ļ′=1√đ‘Ĩ(−5đ‘Ĩ−2)īŽĸīŽĄ
  • Dđ‘Ļ′=−1√đ‘Ĩ(−5đ‘Ĩ−2)īŽĄīŽĸ
  • Eđ‘Ļ′=−1√đ‘Ĩ(−5đ‘Ĩ−2)īŽĸīŽĄ

Q17:

Evaluate ddđ‘Ļđ‘Ĩ at (1,−1) if đ‘Ļ=−2đ‘Ĩ√3đ‘Ĩ+1īŠ¨.

  • A−3
  • B−32
  • C94
  • D−14

Q18:

If đ‘Ļ=đ‘Ĩ√𝑎+5đ‘ĨīŠ¨īŠ¨, find đ‘Ĩī€Ŋđ‘Ļđ‘Ĩī‰īŠ¯dd.

  • A𝑎đ‘Ĩđ‘ĻīŠŦīŠŠ
  • Bđ‘Ĩđ‘ĻīŠ¯īŠŠ
  • Cđ‘ĻīŠŠ
  • D𝑎đ‘Ļđ‘ĨīŠ¨īŠŠīŠŦ
  • Eđ‘Ĩđ‘ĻīŠ­

Q19:

Evaluate ddđ‘Ļđ‘Ĩ at đ‘Ĩ=1 if đ‘Ļ=ī€š6đ‘Ĩ−2đ‘Ĩ−3ī…ī€šđ‘Ĩ−2ī…īŠ¨īŠąīŠŠīŠ¨īŠĢ.

Q20:

Find the derivative of the function 𝑈(đ‘Ļ)=ī€žđ‘Ļ−4đ‘Ļ−4īŠīŠŠīŠŦīŠĒ.

  • A𝑈′(đ‘Ļ)=4ī€šđ‘Ļ−4ī…ī€šâˆ’3đ‘Ļ+24đ‘Ļ−12đ‘Ļī…(đ‘Ļ−4)īŠŠīŠŠīŠŽīŠĢīŠ¨īŠŦīŠĒ
  • B𝑈′(đ‘Ļ)=4ī€šđ‘Ļ−4ī…ī€š9đ‘Ļ−24đ‘Ļ−12đ‘Ļī…(đ‘Ļ−4)īŠŠīŠŠīŠŽīŠĢīŠ¨īŠŦīŠĒ
  • C𝑈′(đ‘Ļ)=4ī€šđ‘Ļ−4ī…ī€šâˆ’3đ‘Ļ+24đ‘Ļ−12đ‘Ļī…(đ‘Ļ−4)īŠŠīŠĒīŠŽīŠĢīŠ¨īŠŦīŠŦ
  • D𝑈′(đ‘Ļ)=4ī€šđ‘Ļ−4ī…ī€šâˆ’3đ‘Ļ+24đ‘Ļ−12đ‘Ļī…(đ‘Ļ−4)īŠŠīŠŠīŠŽīŠĢīŠ¨īŠŦīŠĢ
  • E𝑈′(đ‘Ļ)=4ī€šđ‘Ļ−4ī…ī€š9đ‘Ļ−24đ‘Ļ−12đ‘Ļī…(đ‘Ļ−4)īŠŠīŠŠīŠŽīŠĢīŠ¨īŠŦīŠĢ

Q21:

Determine the derivative of the function 𝑔(đ‘ĸ)=ī€žđ‘ĸ+5đ‘ĸ−1īŠīŠ¨īŠ¨īŠĒ.

  • A𝑔′(đ‘ĸ)=4ī€žđ‘ĸ+5đ‘ĸ−1īŠīŠ¨īŠ¨īŠŠ
  • B𝑔′(đ‘ĸ)=−48đ‘ĸī€šđ‘ĸ+5ī…(đ‘ĸ−1)īŠ¨īŠŠīŠ¨īŠĢ
  • C𝑔′(đ‘ĸ)=−24đ‘ĸī€šđ‘ĸ+5ī…(đ‘ĸ−1)īŠ¨īŠ¨īŠŠīŠ¨īŠĢ
  • D𝑔′(đ‘ĸ)=−48đ‘ĸī€šđ‘ĸ+5ī…(đ‘ĸ−1)īŠ¨īŠŠīŠ¨īŠŠ
  • E𝑔′(đ‘ĸ)=−12đ‘ĸī€šđ‘ĸ+5ī…(đ‘ĸ−1)īŠ¨īŠŠīŠ¨īŠĢ

Q22:

Determine ddđ‘Ļđ‘Ĩ, given that đ‘Ļ=ī€ŧđ‘Ĩ−23đ‘Ĩ+6īˆīŠąīŠĢ.

  • A−145(đ‘Ĩ+6)(đ‘Ĩ−23)īŠĒīŠŦ
  • B−145(đ‘Ĩ+6)(đ‘Ĩ−23)īŠŠīŠĢ
  • C−145(đ‘Ĩ+6)(đ‘Ĩ−23)īŠŦīŠŦ
  • D−17(đ‘Ĩ+6)(đ‘Ĩ−23)īŠĒīŠŦ
  • E85(đ‘Ĩ+6)(đ‘Ĩ−23)īŠĒīŠŦ

Q23:

Given that đ‘Ļ=ī€žđ‘Ĩ−5đ‘Ĩ+5īŠīŠ¨īŠ¨īŠ, determine ddđ‘Ļđ‘Ĩ.

  • A20đ‘Ĩđ‘Ļđ‘Ĩ+25īŠĒ
  • B20𝑛đ‘Ĩđ‘Ļđ‘Ĩ−25īŠĒ
  • C20đ‘Ĩđ‘Ļđ‘Ĩ−25īŠĒ
  • D20𝑛đ‘Ĩđ‘Ļđ‘Ĩ+25īŠĒ

Q24:

Find the derivative of the function đģ(𝑟)=ī€šâˆ’đ‘Ÿ+1ī…(−𝑟+4)īŠ¨īŠĢīŠ­.

  • Ađģ′(𝑟)=ī€šâˆ’đ‘Ÿ+1ī…ī€š3𝑟−40𝑟+7ī…(−𝑟+4)īŠ¨īŠĒīŠ¨īŠŽ
  • Bđģ′(𝑟)=ī€šâˆ’đ‘Ÿ+1ī…ī€š17𝑟−40𝑟−7ī…(−𝑟+4)īŠ¨īŠĒīŠ¨īŠŦ
  • Cđģ′(𝑟)=ī€šâˆ’đ‘Ÿ+1ī…ī€š3𝑟−40𝑟+7ī…(−𝑟+4)īŠ¨īŠĒīŠ¨īŠŦ
  • Dđģ′(𝑟)=ī€šâˆ’đ‘Ÿ+1ī…ī€š17𝑟−40𝑟−7ī…(−𝑟+4)īŠ¨īŠĒīŠ¨īŠŽ
  • Eđģ′(𝑟)=10𝑟ī€šâˆ’đ‘Ÿ+1ī…7(−𝑟+4)īŠ¨īŠĒīŠŽ

Q25:

If đ‘Ļ=ī€šđ‘Ĩ+4ī…(−4đ‘Ĩ−3)īŠ¨īŠĢīŠ¨īŠĒ, find ddđ‘Ļđ‘Ĩ.

  • A−2đ‘Ĩī€šđ‘Ĩ+4ī…(4đ‘Ĩ+3)ī€š36đ‘Ĩ+79ī…īŠ¨īŠĒīŠ¨īŠĢīŠ¨
  • B2đ‘Ĩī€šđ‘Ĩ+4ī…(4đ‘Ĩ+3)ī€š4đ‘Ĩ−49ī…īŠ¨īŠĒīŠ¨īŠĢīŠ¨
  • C2đ‘Ĩī€šđ‘Ĩ+4ī…4đ‘Ĩ+3ī€š36đ‘Ĩ+79ī…īŠ¨īŠĒīŠ¨īŠ¨
  • D2đ‘Ĩī€šđ‘Ĩ+4ī…(4đ‘Ĩ+3)ī€š4đ‘Ĩ−49ī…īŠ¨īŠĒīŠ¨īŠ¨īŠ¨

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.