Worksheet: Power Rule of Derivatives

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In this worksheet, we will practice using the power rule of derivatives and the derivative of a sum of functions to find the derivatives of polynomials and general power functions.

Q1:

Differentiate the function 𝑓(𝑥)=−𝑥7−3𝑥−4.

  • A 𝑓 ′ ( 𝑥 ) = − 𝑥 7 − 3 𝑥 − 4 𝑥
  • B 𝑓 ′ ( 𝑥 ) = − 2 𝑥 7 − 3
  • C 𝑓 ′ ( 𝑥 ) = − 𝑥 7 − 3
  • D 𝑓 ′ ( 𝑥 ) = − 2 𝑥 7 − 3 𝑥 − 4 𝑥
  • E 𝑓 ′ ( 𝑥 ) = − 2 𝑥 7 − 3 𝑥 − 4

Q2:

Given that 𝑦=𝑥+𝑥+8𝑥 and 𝑧=𝑥(𝑥−4)(𝑥−1), determine dddd𝑦𝑥−𝑧𝑥.

  • A − 4 𝑥 − 4
  • B 1 2 𝑥 + 4
  • C − 2 𝑥 − 4
  • D 1 2 𝑥 + 1 2
  • E − 8 𝑥 + 4

Q3:

Find dd𝑦𝑥, given that −6𝑥𝑦=11.

  • A − 1 1 𝑦 6 𝑥
  • B 1 1 6 𝑥
  • C 1 1 6 𝑥
  • D 1 1 − 6 𝑦 𝑥 𝑥

Q4:

Given that 𝑦=15−1𝑥+13𝑥, determine 𝑦′.

  • A 6 𝑥 + 2 0 3 𝑥
  • B 5 𝑥 + 1 9 3 𝑥
  • C 6 𝑥 + 2 0 3 𝑥
  • D 1 𝑥 + 1 3 𝑥

Q5:

Find dd𝑦𝑥, given that 𝑦=7𝑥+1𝑥.

  • A 3 5 𝑥 − 6 𝑥
  • B 3 5 𝑥 − 6 𝑥
  • C 7 𝑥 + 1 𝑥
  • D 3 5 𝑥 − 6 𝑥
  • E 3 5 𝑥 + 6 𝑥

Q6:

Find dd𝑦𝑥, given that 𝑦=22𝑥.

  • A 2 2 𝑥
  • B 8 8 𝑥
  • C 8 8 𝑥
  • D 8 8 𝑥

Q7:

Determine the first derivative of the function 𝑦=2𝑥9𝑥−3𝑥+10𝑥.

  • A 5 4 𝑥 − 1 2 𝑥 + 1 0 𝑥
  • B 5 4 𝑥 − 1 2 𝑥 + 1 0
  • C 1 8 𝑥 − 6 𝑥 + 1 0 𝑥
  • D 1 8 𝑥 − 6 𝑥 + 1 0

Q8:

Evaluate 𝑓′(3), where 𝑓(𝑥)=17𝑥+52𝑥.

Q9:

Differentiate the function 𝑓(𝑡)=6.3𝑡+5.2𝑡+7.3.

  • A 𝑓 ′ ( 𝑡 ) = 6 . 3 𝑡 + 5 . 2 𝑡
  • B 𝑓 ′ ( 𝑡 ) = 2 5 . 2 𝑡 + 5 . 2 𝑡
  • C 𝑓 ′ ( 𝑡 ) = 3 1 . 5 𝑡 + 1 0 . 4 𝑡
  • D 𝑓 ′ ( 𝑡 ) = 3 1 . 5 𝑡 + 1 0 . 4 𝑡 + 7 . 3 𝑡
  • E 𝑓 ′ ( 𝑡 ) = 6 . 3 𝑡 + 5 . 2 𝑡 + 7 . 3 𝑡

Q10:

Given that 𝑦=3𝑥𝑒, determine dd𝑦𝑥.

  • A 2 1 𝑥 𝑒
  • B − 1 8 𝑥 𝑒 + 2 1 𝑥 𝑒
  • C − 1 8 𝑥 𝑒 + 3 𝑥 𝑒
  • D 3 𝑥 𝑒

Q11:

Find dd𝑦𝑥 if 𝑦=2𝑥.

  • A − 1 6 𝑥
  • B − 1 4 𝑥
  • C 2 𝑥
  • D − 1 4 𝑥

Q12:

Given that 𝑦=−25𝑥, determine dd𝑦𝑥.

Q13:

Evaluate dd𝑥√3𝑥+𝑥9+6𝜋.

  • A 7 √ 3 𝑥 + 8 𝑥
  • B 7 √ 3 𝑥 + 8 9 𝑥
  • C √ 3 7 𝑥 + 𝑥 8 1
  • D 7 √ 3 𝑥 + 𝑥
  • E 7 √ 3 𝑥 + 𝑥 + 6

Q14:

Find the first derivative of the function 𝑓(𝑥)=−2𝑥+10.

Q15:

Find the first derivative of the function 𝑦=𝑥−𝑥4−3𝑥+6.

  • A 𝑥 − 𝑥 4 − 3
  • B 4 𝑥 − 3 𝑥 4 − 3 𝑥 + 6 𝑥
  • C 3 𝑥 − 𝑥 2 − 3
  • D 4 𝑥 − 3 𝑥 4 − 3

Q16:

Find the first derivative of the function 𝑦=𝑥+83𝑥−8𝑥+6.

  • A 3 𝑥 + 1 6 𝑥 + 6 𝑥 − 6 4
  • B 1 8 𝑥 + 4 8 𝑥 + 1 2 𝑥 − 6 4
  • C 1 5 𝑥 + 4 8 𝑥 + 1 2 𝑥 − 6 4
  • D 1 2 𝑥 + 3 2 𝑥 + 6 𝑥 − 6 4

Q17:

Find dd𝑦𝑥, given that 𝑦=−8𝑥−4𝑥+3𝑥+5.

  • A − 8 𝑥 − 4 𝑥 + 3
  • B − 7 2 𝑥 − 2 0 𝑥
  • C − 7 2 𝑥 − 2 0 𝑥 + 3
  • D − 7 2 𝑥 − 2 0 𝑥 + 3

Q18:

Evaluate dd𝑥−3𝑥−4𝑥+5.

  • A − 3 𝑥 − 4
  • B − 6 𝑥 − 4
  • C − 9 𝑥 − 4
  • D − 9 𝑥 − 4 𝑥

Q19:

Given that 𝑓(𝑥)=−𝑥+𝑚𝑥+1, determine 𝑚 if 𝑓′(3)=1.

Q20:

If the function 𝑓(𝑥)=−3𝑥−5, find lim→𝑓(𝑥+ℎ)−𝑓(𝑥)ℎ.

  • A − 2 7 𝑥
  • B − 2 7 𝑥 − 5 ℎ
  • C − 2 7
  • D − 2 7 𝑥
  • E − 3 𝑥

Q21:

Find dd𝑦𝑥 if 𝑦=6√𝑥7 .

  • A 6 7 √ 𝑥
  • B 3 √ 𝑥 7
  • C 3 7 √ 𝑥
  • D − 3 7 √ 𝑥

Q22:

If 𝑦=5(2𝑥+2𝑥)cossin, find dd𝑦𝑥.

Q23:

If 𝑦=158𝑥−158𝑥sincos, find dd𝑦𝑥.

  • A0
  • B 1 5 1 6 𝑥 s i n
  • C − 1 5 1 6 𝑥 s i n
  • D − 2 4 0 1 6 𝑥 s i n
  • E 2 4 0 1 6 𝑥 s i n

Q24:

Find dd𝑦𝑥, given that 𝑦=−43𝑥.

  • A − 3 4 4 𝑥
  • B − 3 4 4 𝑥
  • C 3 4 4 𝑥
  • D − 4 3 𝑥

Q25:

Find dd𝑦𝑥, given that 𝑦=3𝑥+4𝑥+6−7𝑥−8𝑥.

  • A 1 2 𝑥 + 8 𝑥 + 6 + 4 9 𝑥 + 6 4 𝑥
  • B 1 2 𝑥 + 8 𝑥 + 4 9 𝑥 + 6 4 𝑥
  • C 1 2 𝑥 + 8 𝑥 + 6 + 4 9 𝑥 + 6 4 𝑥
  • D 3 𝑥 + 4 𝑥 + 7 𝑥 + 8 𝑥
  • E 9 𝑥 + 4 𝑥 + 5 6 𝑥 + 7 2 𝑥

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