Worksheet: Power Rule of Derivatives

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 ) = 𝑥 7 3 𝑥 4 2 .

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

Q2:

Given that 𝑦 = 𝑥 + 𝑥 + 8 𝑥 3 2 and 𝑧 = 𝑥 ( 𝑥 4 ) ( 𝑥 1 ) , determine d d d d 𝑦 𝑥 𝑧 𝑥 .

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

Q3:

Find d d 𝑦 𝑥 , given that 6 𝑥 𝑦 = 1 1 .

  • A 1 1 6 𝑥
  • B 1 1 𝑦 6 𝑥
  • C 1 1 6 𝑦 𝑥 𝑥
  • D 1 1 6 𝑥 2

Q4:

Given that 𝑦 = 1 5 1 𝑥 + 1 3 𝑥 6 2 0 , determine 𝑦 .

  • A 1 𝑥 + 1 3 𝑥 7 1 9
  • B 6 𝑥 + 2 0 3 𝑥 6 2 0
  • C 5 𝑥 + 1 9 3 𝑥 7 1 9
  • D 6 𝑥 + 2 0 3 𝑥 7 1 9

Q5:

Find d d 𝑦 𝑥 , given that 𝑦 = 7 𝑥 + 1 𝑥 5 6 .

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

Q6:

Find d d 𝑦 𝑥 , given that 𝑦 = 2 2 𝑥 4 .

  • A 8 8 𝑥 5
  • B 8 8 𝑥 4
  • C 2 2 𝑥 5
  • D 8 8 𝑥 3

Q7:

Determine the first derivative of the function 𝑦 = 2 𝑥 9 𝑥 3 𝑥 + 1 0 𝑥 2 .

  • A 5 4 𝑥 1 2 𝑥 + 1 0 𝑥 4 3 2
  • B 1 8 𝑥 6 𝑥 + 1 0 2
  • C 1 8 𝑥 6 𝑥 + 1 0 𝑥 4 3 2
  • D 5 4 𝑥 1 2 𝑥 + 1 0 2

Q8:

Evaluate 𝑓 ( 3 ) , where 𝑓 ( 𝑥 ) = 1 7 𝑥 + 5 2 𝑥 2 .

Q9:

Differentiate the function 𝑓 ( 𝑡 ) = 6 . 3 𝑡 + 5 . 2 𝑡 + 7 . 3 5 2 .

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

Q10:

Given that 𝑦 = 3 𝑥 𝑒 7 6 , determine d d 𝑦 𝑥 .

  • A 1 8 𝑥 𝑒 + 2 1 𝑥 𝑒 7 6 6 6
  • B 3 𝑥 𝑒 6 6
  • C 1 8 𝑥 𝑒 + 3 𝑥 𝑒 7 6 6 6
  • D 2 1 𝑥 𝑒 6 6

Q11:

Find d d 𝑦 𝑥 if 𝑦 = 2 𝑥 7 .

  • A 2 𝑥 8
  • B 1 4 𝑥 6
  • C 1 6 𝑥 8
  • D 1 4 𝑥 8

Q12:

Given that 𝑦 = 2 5 𝑥 , determine d d 𝑦 𝑥 .

Q13:

Given that 𝑦 = ( 𝑘 5 𝑥 ) 3 , and d d 𝑦 𝑥 = 1 5 at 𝑥 = 2 , find the values of 𝑘 .

  • A 1 5 , 2
  • B15, 5
  • C 1 0 , 1 2
  • D 9 , 1 1

Q14:

Given that 𝑦 = ( 𝑥 1 1 ) 3 , determine 𝑦 𝑥 𝑦 ( 𝑥 1 1 ) 𝑦 𝑥 d d d d 3 .

  • A ( 𝑥 1 1 ) ( 𝑥 1 1 ) 2 4 6
  • B 9 0 𝑦 2
  • C ( 𝑥 1 1 ) ( 𝑥 1 1 ) 3 0 8
  • D 2 4 𝑦 2

Q15:

Evaluate d d 𝑥 3 𝑥 + 𝑥 9 + 6 𝜋 7 9 .

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

Q16:

Solve 𝑓 ( 𝑥 ) = 8 1 for 𝑥 , where 𝑓 ( 𝑥 ) = ( 9 𝑥 + 1 ) 9 .

  • A 𝑥 = 8 1 0 o r
  • B 𝑥 = 2 9 0 o r
  • C 𝑥 = 1 0 8 o r
  • D 𝑥 = 0 2 9 o r

Q17:

Find the first derivative of the function 𝑓 ( 𝑥 ) = 2 𝑥 + 1 0 .

Q18:

Find d d 𝑦 𝑥 , given that 𝑦 = ( 7 𝑥 + 3 ) 4 .

  • A 4 ( 7 𝑥 + 3 ) 3
  • B 2 1 ( 7 𝑥 + 3 ) 3
  • C 2 8 ( 7 𝑥 + 3 ) 5
  • D 2 8 ( 7 𝑥 + 3 ) 3

Q19:

Find the first derivative of the function 𝑦 = 𝑥 𝑥 4 3 𝑥 + 6 4 3 .

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

Q20:

Find the first derivative of the function 𝑦 = 𝑥 + 8 3 𝑥 8 𝑥 + 6 2 3 .

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

Q21:

Solve 𝑓 ( 𝑥 ) = 2 7 for 𝑥 , where 𝑓 ( 𝑥 ) = ( 9 𝑥 4 ) 3 .

  • A 𝑥 = 1 3 , 𝑥 = 1 3
  • B 𝑥 = 5 9 , 𝑥 = 5 9
  • C 𝑥 = 5 9
  • D 𝑥 = 1 3 , 𝑥 = 5 9
  • E 𝑥 = 1 3

Q22:

Find d d 𝑦 𝑥 , given that 𝑦 = 8 𝑥 4 𝑥 + 3 𝑥 + 5 9 5 .

  • A 7 2 𝑥 2 0 𝑥 + 3 9 5
  • B 7 2 𝑥 2 0 𝑥 1 0 6
  • C 8 𝑥 4 𝑥 + 3 8 4
  • D 7 2 𝑥 2 0 𝑥 + 3 8 4

Q23:

Suppose that 𝑓 ( 𝑥 ) = ( 2 𝑥 + 𝑎 ) 3 𝑥 𝑎 2 and 𝑓 ( 1 ) = 1 0 . Determine 𝑎 .

Q24:

Evaluate d d 𝑥 3 𝑥 4 𝑥 + 5 2 .

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

Q25:

Given that 𝑓 ( 𝑥 ) = 𝑥 + 𝑚 𝑥 + 1 2 , determine 𝑚 if 𝑓 ( 3 ) = 1 .

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