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Lesson: Rules of Differentiation

Sample Question Videos

Worksheet • 25 Questions • 2 Videos

Q1:

Find , given that .

  • A
  • B
  • C
  • D

Q2:

Find d d 𝑦 π‘₯ , given that 𝑦 = 3 π‘₯ + 4 π‘₯ + 6 βˆ’ 7 π‘₯ βˆ’ 8 π‘₯ 4 2 7 8 .

  • A 1 2 π‘₯ + 8 π‘₯ + 4 9 π‘₯ + 6 4 π‘₯ 3 8 9
  • B 9 π‘₯ + 4 π‘₯ + 5 6 π‘₯ + 7 2 π‘₯ 3 8 9
  • C 1 2 π‘₯ + 8 π‘₯ + 6 + 4 9 π‘₯ + 6 4 π‘₯ 3 8 9
  • D 3 π‘₯ + 4 π‘₯ + 7 π‘₯ + 8 π‘₯ 3 8 9
  • E 1 2 π‘₯ + 8 π‘₯ + 6 + 4 9 π‘₯ + 6 4 π‘₯ 3 6 7

Q3:

Differentiate 𝑓 ( π‘₯ ) = βˆ’ 5 π‘Ž π‘₯ βˆ’ 9 𝑏 2 , where π‘Ž and 𝑏 are two constants.

  • A βˆ’ 1 0 π‘Ž π‘₯
  • B 1 0 π‘Ž π‘₯
  • C 5 π‘Ž π‘₯
  • D βˆ’ 5 π‘Ž π‘₯

Q4:

Differentiate 𝑓 ( π‘₯ ) = 2 π‘Ž π‘₯ + 𝑏 2 , where π‘Ž and 𝑏 are two constants.

  • A 4 π‘Ž π‘₯
  • B βˆ’ 4 π‘Ž π‘₯
  • C βˆ’ 2 π‘Ž π‘₯
  • D 2 π‘Ž π‘₯

Q5:

If 𝑦 = π‘₯ βˆ’ 7 √ π‘₯ οŠͺ  , find d d 𝑦 π‘₯ .

  • A 4 π‘₯ βˆ’ 4 9 √ π‘₯ 2  
  • B 3 π‘₯ βˆ’ 3 5 √ π‘₯ 2  
  • C 4 π‘₯ + 7 √ π‘₯ 2  
  • D π‘₯ βˆ’ 7 √ π‘₯  

Q6:

Find the first derivative of the function 𝑦 = √ π‘₯ + 7 √ π‘₯ 5 5 .

  • A 5 2 π‘₯ βˆ’ 3 5 2 π‘₯ 3 2 7 2 βˆ’
  • B π‘₯ βˆ’ 7 π‘₯ 3 2 7 2 βˆ’
  • C 3 2 π‘₯ βˆ’ 2 1 2 π‘₯ 3 2 7 2 βˆ’
  • D 5 2 π‘₯ + 3 5 2 π‘₯ 3 2 7 2 βˆ’

Q7:

Find the first derivative of the function 𝑦 = ο€Ή 3 π‘₯ + 7  ο€Ή 7 βˆ’ 3 π‘₯  5 5 .

  • A βˆ’ 9 0 π‘₯ 9
  • B βˆ’ 9 0 π‘₯ 1 0
  • C 9 0 π‘₯ 9
  • D βˆ’ 1 8 π‘₯ 1 0
  • E 9 0 π‘₯ 1 0

Q8:

Find the first derivative of the function 𝑦 = ( 5 π‘₯ + 2 ) ( 9 π‘₯ + 6 π‘₯ + 4 ) 2 3 .

  • A 2 2 5 π‘₯ + 1 4 4 π‘₯ + 4 0 π‘₯ + 1 2 4 2
  • B 2 2 5 π‘₯ + 1 4 4 π‘₯ + 4 0 π‘₯ 5 3 2
  • C 2 2 5 π‘₯ + 1 4 4 π‘₯ + 4 0 π‘₯ 6 4 3
  • D 4 5 π‘₯ + 4 8 π‘₯ + 2 0 π‘₯ 4 2

Q9:

Find the first derivative of the function 𝑦 = 9 π‘₯ + 5 π‘₯ ο€Ό 4 π‘₯ + 5 π‘₯  2 2 .

  • A 4 0 0 π‘₯ + 4 0 0 π‘₯ βˆ’ 1 2 5 π‘₯ + 9 4 βˆ’ 2
  • B 8 0 π‘₯ + 2 0 0 π‘₯ + 1 2 5 π‘₯ + 9 4 βˆ’ 2
  • C 3 2 0 π‘₯ + 2 0 0 π‘₯ βˆ’ 2 5 0 π‘₯ + 9 4 βˆ’ 2
  • D 4 0 0 π‘₯ + 4 0 0 π‘₯ + 1 2 5 π‘₯ + 9 4 βˆ’ 2

Q10:

Evaluate d d π‘₯ ο€Ώ βˆ’ 5 √ π‘₯  3 .

  • A 5 3 √ π‘₯ 3 4
  • B 5 √ π‘₯ 3 3 2
  • C βˆ’ 5 √ π‘₯ 3 4
  • D βˆ’ 5 2 √ π‘₯ 3 4

Q11:

Differentiate 𝑓 ( π‘₯ ) = 4 √ π‘₯ + 8 , and identify the value of π‘₯ at which the function is NOT differentiable.

  • A 𝑓 β€² ( π‘₯ ) = 2 √ π‘₯ + 8 , the function is not differentiable at π‘₯ β‰₯ βˆ’ 8 .
  • B 𝑓 β€² ( π‘₯ ) = 4 √ π‘₯ + 8 , the function is not differentiable at π‘₯ β‰₯ 8 .
  • C 𝑓 β€² ( π‘₯ ) = 4 √ π‘₯ + 8 , the function is not differentiable at π‘₯ β‰₯ βˆ’ 8 .
  • D 𝑓 β€² ( π‘₯ ) = 2 √ π‘₯ + 8 , the function is not differentiable at π‘₯ β‰₯ 8 .

Q12:

Find , given that .

  • A
  • B
  • C
  • D

Q13:

Find the first derivative of 𝑦 = 9 π‘₯ βˆ’ 7 √ π‘₯ 6 with respect to π‘₯ .

  • A 5 4 π‘₯ βˆ’ 7 2 √ π‘₯ 5
  • B 5 4 π‘₯ βˆ’ 7 π‘₯ 5
  • C 5 4 π‘₯ βˆ’ 7 √ π‘₯ 5
  • D 5 4 π‘₯ βˆ’ 7 2 π‘₯ 5
  • E 5 4 π‘₯ βˆ’ 7 5

Q14:

Evaluate d d π‘₯  βˆ’ 5 π‘₯  1 9 .

  • A βˆ’ 5 9 √ π‘₯ 9 8
  • B βˆ’ 5 √ π‘₯ 9 9 1 0
  • C βˆ’ 5 √ π‘₯ 9 8
  • D βˆ’ 5 8 √ π‘₯ 9 8

Q15:

Differentiate 𝑓 ( π‘₯ ) = 9 π‘₯ + 3 √ π‘₯ βˆ’ 5 π‘₯ βˆ’ 6 4 9 2 .

  • A 3 6 π‘₯ + 3 2 √ π‘₯ βˆ’ 4 5 2 π‘₯ 3 7 2
  • B 9 π‘₯ + 3 √ π‘₯ βˆ’ 5 π‘₯ 3 7 2
  • C 2 7 π‘₯ βˆ’ 3 √ π‘₯ βˆ’ 4 5 2 π‘₯ 3 9 2
  • D 3 6 π‘₯ + 3 2 √ π‘₯ βˆ’ 5 π‘₯ 4 9 2

Q16:

If 𝑦 = √ 2 βˆ’ 3 π‘₯ , which of the following is the same as d d 𝑦 π‘₯ ?

  • A βˆ’ 3 2 𝑦
  • B βˆ’ 2 𝑦 3
  • C 2 𝑦 3
  • D 3 2 𝑦

Q17:

Find d d π‘₯ ο€Ί βˆ’ 2 √ π‘₯ βˆ’ 7 π‘₯  .

  • A βˆ’ 7 βˆ’ 1 √ π‘₯
  • B βˆ’ 7 βˆ’ 2 √ π‘₯
  • C βˆ’ √ π‘₯ βˆ’ 7
  • D βˆ’ 7 + 1 √ π‘₯

Q18:

Find the first derivative of the function 𝑦 = 9 π‘₯ + 2 π‘₯ + 4 √ π‘₯ π‘₯ 2 .

  • A 9 βˆ’ 2 π‘₯ βˆ’ 3 2
  • B 9 βˆ’ 6 π‘₯ βˆ’ 3 2
  • C 9 βˆ’ 4 π‘₯ βˆ’ 3 2
  • D 9 βˆ’ 2 π‘₯ 1 2

Q19:

Find d d 𝑦 π‘₯ , given that 𝑦 = βˆ’ π‘₯ 4 βˆ’ 5 + 5 π‘₯ 5 2 .

  • A βˆ’ 5 π‘₯ 4 βˆ’ 1 0 π‘₯ 4 3
  • B βˆ’ 3 π‘₯ 2 + 1 0 π‘₯ 6
  • C βˆ’ π‘₯ 4 βˆ’ 5 π‘₯ 4 3
  • D βˆ’ 5 π‘₯ 4 + 1 0 π‘₯ 5 2
  • E βˆ’ 5 π‘₯ 4 βˆ’ 1 0 π‘₯ 4

Q20:

Differentiate 𝑦 = √ π‘₯ ο€Ώ 5 π‘₯ √ π‘₯ βˆ’ 4 π‘₯ √ π‘₯ βˆ’ 1  .

  • A 1 0 π‘₯ βˆ’ 4 βˆ’ 1 2 √ π‘₯
  • B 5 π‘₯ βˆ’ 4 βˆ’ 1 √ π‘₯
  • C 5 π‘₯ βˆ’ 4 π‘₯ βˆ’ 1 √ π‘₯ 2
  • D 1 0 π‘₯ βˆ’ 4 π‘₯ βˆ’ 1 2 √ π‘₯ 3 2 3

Q21:

Find d d 𝑦 π‘₯ , given that 𝑦 = 5 π‘₯ + 3 π‘₯ √ π‘₯ + √ 2 1 π‘₯ + 1 7 .

  • A βˆ’ 5 π‘₯ + 9 2 √ π‘₯ + √ 2 1 2
  • B βˆ’ 5 + 9 2 √ π‘₯ + √ 2 1 π‘₯ 5 2
  • C 5 π‘₯ + 9 2 √ π‘₯ + √ 2 1 π‘₯ 3
  • D 5 π‘₯ + 3 √ π‘₯ + √ 2 1 2

Q22:

Differentiate 𝐺 ( 𝑑 ) = √ 5 𝑑 + √ 2 2 𝑑 .

  • A 𝐺 β€² ( 𝑑 ) = βˆ’ √ 2 2 𝑑 + √ 5 2 √ 𝑑 2
  • B 𝐺 β€² ( 𝑑 ) = βˆ’ √ 2 2 𝑑 + √ 5 1 0 √ 𝑑 2
  • C 𝐺 β€² ( 𝑑 ) = √ 2 2 𝑑 + √ 5 1 0 √ 𝑑 2
  • D 𝐺 β€² ( 𝑑 ) = √ 2 2 𝑑 + √ 5 2 √ 𝑑 2
  • E 𝐺 β€² ( 𝑑 ) = √ 2 2 𝑑 + √ 5 5 √ 𝑑 2

Q23:

Find the first derivative of the function 𝑦 = 1 2 π‘₯ + 1 .

  • A βˆ’ 2 ( 2 π‘₯ + 1 ) 2
  • B 2 ( 2 π‘₯ + 1 ) 2
  • C βˆ’ 1 ( 2 π‘₯ + 1 ) 2
  • D 1 ( 2 π‘₯ + 1 ) 2

Q24:

Let 𝑔 ( π‘₯ ) = βˆ’ 2 𝑓 ( π‘₯ ) + 5 β„Ž ( π‘₯ ) . If 𝑓 β€² ( βˆ’ 8 ) = 9 and β„Ž β€² ( βˆ’ 8 ) = βˆ’ 1 , find 𝑔 β€² ( βˆ’ 8 ) .

Q25:

Differentiate 𝑦 = √ π‘₯ ( βˆ’ 2 π‘₯ + 1 ) 3 .

  • A 𝑦 β€² = βˆ’ 8 √ π‘₯ 3 + 1 3 π‘₯ 3 2 3
  • B 𝑦 β€² = βˆ’ 2 √ π‘₯ 3 βˆ’ 2 3 π‘₯ 3 2 3
  • C 𝑦 β€² = βˆ’ 2 √ π‘₯ + 1 π‘₯ 3 2 3
  • D 𝑦 β€² = βˆ’ 2 π‘₯ + π‘₯ 7 3 4 3
  • E 𝑦 β€² = βˆ’ 8 π‘₯ 3 + π‘₯ 3 7 3 4 3
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