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.

  • A𝑓′(𝑥)=−𝑥7−3𝑥−4𝑥
  • B𝑓′(𝑥)=−2𝑥7−3
  • C𝑓′(𝑥)=−𝑥7−3
  • D𝑓′(𝑥)=−2𝑥7−3𝑥−4𝑥
  • E𝑓′(𝑥)=−2𝑥7−3𝑥−4

Q2:

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

  • A−4𝑥−4
  • B12𝑥+4
  • C12𝑥+12
  • D−2𝑥−4
  • E−8𝑥+4

Q3:

Find dd𝑦𝑥, given that −6𝑥𝑦=11.

  • A−11𝑦6𝑥
  • B116𝑥
  • C116𝑥
  • D11−6𝑦𝑥𝑥

Q4:

Given that 𝑦=15−1𝑥+13𝑥, determine 𝑦′.

  • A1𝑥+13𝑥
  • B6𝑥+203𝑥
  • C6𝑥+203𝑥
  • D5𝑥+193𝑥

Q5:

Find dd𝑦𝑥, given that 𝑦=7𝑥+1𝑥.

  • A35𝑥−6𝑥
  • B35𝑥−6𝑥
  • C35𝑥+6𝑥
  • D35𝑥−6𝑥
  • E7𝑥+1𝑥

Q6:

Find dd𝑦𝑥, given that 𝑦=22𝑥.

  • A88𝑥
  • B88𝑥
  • C22𝑥
  • D88𝑥

Q7:

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

  • A54𝑥−12𝑥+10
  • B18𝑥−6𝑥+10
  • C54𝑥−12𝑥+10𝑥
  • D18𝑥−6𝑥+10𝑥

Q8:

Evaluate 𝑓′(3), where 𝑓(𝑥)=17𝑥+52𝑥.

Q9:

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

  • A𝑓′(𝑡)=6.3𝑡+5.2𝑡
  • B𝑓′(𝑡)=25.2𝑡+5.2𝑡
  • C𝑓′(𝑡)=31.5𝑡+10.4𝑡+7.3𝑡
  • D𝑓′(𝑡)=6.3𝑡+5.2𝑡+7.3𝑡
  • E𝑓′(𝑡)=31.5𝑡+10.4𝑡

Q10:

Given that 𝑦=3𝑥𝑒, determine dd𝑦𝑥.

  • A21𝑥𝑒
  • B3𝑥𝑒
  • C−18𝑥𝑒+21𝑥𝑒
  • D−18𝑥𝑒+3𝑥𝑒

Q11:

Find dd𝑦𝑥 if 𝑦=2𝑥.

  • A2𝑥
  • B−16𝑥
  • C−14𝑥
  • D−14𝑥

Q12:

Given that 𝑦=−25𝑥, determine dd𝑦𝑥.

Q13:

Evaluate dd𝑥√3𝑥+𝑥9+6𝜋.

  • A7√3𝑥+8𝑥
  • B7√3𝑥+89𝑥
  • C7√3𝑥+𝑥
  • D√37𝑥+𝑥81
  • E7√3𝑥+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.

  • A4𝑥−3𝑥4−3
  • B3𝑥−𝑥2−3
  • C𝑥−𝑥4−3
  • D4𝑥−3𝑥4−3𝑥+6𝑥

Q16:

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

  • A3𝑥+16𝑥+6𝑥−64
  • B12𝑥+32𝑥+6𝑥−64
  • C15𝑥+48𝑥+12𝑥−64
  • D18𝑥+48𝑥+12𝑥−64

Q17:

Find dd𝑦𝑥, given that 𝑦=−8𝑥−4𝑥+3𝑥+5.

  • A−72𝑥−20𝑥+3
  • B−72𝑥−20𝑥+3
  • C−8𝑥−4𝑥+3
  • D−72𝑥−20𝑥

Q18:

Evaluate dd𝑥−3𝑥−4𝑥+5.

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

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−27𝑥
  • B−3𝑥
  • C−27
  • D−27𝑥
  • E−27𝑥−5ℎ

Q21:

Find dd𝑦𝑥 if 𝑦=6√𝑥7 .

  • A37√𝑥
  • B−37√𝑥
  • C67√𝑥
  • D3√𝑥7

Q22:

If 𝑦=5(2𝑥+2𝑥)cossin, find dd𝑦𝑥.

Q23:

If 𝑦=158𝑥−158𝑥sincos, find dd𝑦𝑥.

  • A24016𝑥sin
  • B−24016𝑥sin
  • C1516𝑥sin
  • D0
  • E−1516𝑥sin

Q24:

Find dd𝑦𝑥, given that 𝑦=−43𝑥.

  • A−43𝑥
  • B344𝑥
  • C−344𝑥
  • D−344𝑥

Q25:

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

  • A12𝑥+8𝑥+6+49𝑥+64𝑥
  • B9𝑥+4𝑥+56𝑥+72𝑥
  • C12𝑥+8𝑥+6+49𝑥+64𝑥
  • D3𝑥+4𝑥+7𝑥+8𝑥
  • E12𝑥+8𝑥+49𝑥+64𝑥

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