Worksheet: Differentiation of Logarithmic Functions

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In this worksheet, we will practice finding the derivatives of logarithmic functions.

Q1:

Find dd𝑦𝑥, given that 𝑦=(4𝑥+5)ln.

  • A284𝑥+5
  • B28(4𝑥+5)
  • C44𝑥+5
  • D1(4𝑥+5)

Q2:

Determine dd𝑦𝑥, given that 𝑦𝑦=−14𝑥−7:ln.

  • A20𝑥(4𝑥−7)
  • B20𝑥4𝑥−7
  • C−20𝑥4𝑥−7
  • D−20𝑥(4𝑥−7)

Q3:

Find dd𝑦𝑥, given that 𝑦=−8𝑥7𝑥−3ln.

  • A15𝑥−37𝑥−3𝑥
  • B7𝑥−3𝑥7𝑥−6
  • C7𝑥−67𝑥−3𝑥
  • D7𝑥−3𝑥15𝑥−3

Q4:

Find dd𝑦𝑥, given that 𝑦=3𝑥3𝑥ln.

  • A9𝑥+18𝑥3𝑥ln
  • B9𝑥+18𝑥3𝑥ln
  • C3𝑥+18𝑥3𝑥ln
  • D3𝑥+18𝑥3𝑥ln

Q5:

Differentiate 𝐹(𝑡)=−4(𝑡)2𝑡lnsin.

  • A𝐹′(𝑡)=8𝑡(𝑡𝑡2𝑡+2𝑡)𝑡lncossinln
  • B𝐹′(𝑡)=−8𝑡(𝑡𝑡2𝑡+2𝑡)𝑡lncossinln
  • C𝐹′(𝑡)=8(𝑡2𝑡+2𝑡)𝑡lncossinln
  • D𝐹′(𝑡)=8𝑡(𝑡𝑡2𝑡−2𝑡)𝑡lncossinln
  • E𝐹′(𝑡)=−8(𝑡2𝑡+2𝑡)𝑡lncossinln

Q6:

Find dd𝑦𝑥, given that 𝑦=9𝑥9𝑥ln.

  • A9(1−9𝑥)9𝑥lnln
  • B9(9𝑥−𝑥)9𝑥lnln
  • C9(𝑥−9𝑥)9𝑥lnln
  • D9(9𝑥−1)9𝑥lnln

Q7:

If 𝑓(𝑥)=3(2𝑥+4𝑥)lnln, find 𝑓′(1).

  • A19
  • B1
  • C3
  • D9
  • E12

Q8:

Differentiate the function 𝐻(𝑧)=𝑎−𝑧𝑎+𝑧ln.

  • A𝐻′(𝑧)=−2𝑎𝑧𝑧−𝑎
  • B𝐻′(𝑧)=2𝑎𝑧𝑧−𝑎
  • C𝐻′(𝑧)=2𝑎𝑧𝑧−𝑎
  • D𝐻′(𝑧)=𝑧−𝑎2𝑎𝑧
  • E𝐻′(𝑧)=−2𝑎𝑧𝑧−𝑎

Q9:

Differentiate 𝑔(𝑡)=√4𝑡−9ln.

  • A𝑔′(𝑡)=−4𝑡√4𝑡−9ln
  • B𝑔′(𝑡)=2𝑡√4𝑡−9ln
  • C𝑔′(𝑡)=4𝑡√4𝑡−9ln
  • D𝑔′(𝑡)=−2𝑡√4𝑡−9ln

Q10:

Differentiate 𝑓(𝑥)=52𝑥lnsin.

  • A𝑓′(𝑥)=5𝑥cot
  • B𝑓′(𝑥)=10𝑥tan
  • C𝑓′(𝑥)=10𝑥cot
  • D𝑓′(𝑥)=52𝑥tan
  • E𝑓′(𝑥)=−10𝑥cot

Q11:

Differentiate 𝑓(𝑥)=5(5𝑥)sinln.

  • A𝑓′(𝑥)=25(5𝑥)cosln
  • B𝑓′(𝑥)=−25𝑥(5𝑥)cosln
  • C𝑓′(𝑥)=−25(5𝑥)cosln
  • D𝑓′(𝑥)=25𝑥(𝑥)cosln
  • E𝑓′(𝑥)=25𝑥(5𝑥)cosln

Q12:

Given that 𝑓(𝑥)=𝑥cosln, determine 𝑓′(1).

Q13:

Differentiate the function 𝑦=[(𝑎𝑥+𝑏)]tanln.

  • A𝑎((𝑎𝑥+𝑏))𝑎𝑥+𝑏secln
  • Bsecln((𝑎𝑥+𝑏))
  • C−((𝑎𝑥+𝑏))secln
  • D−𝑎((𝑎𝑥+𝑏))𝑎𝑥+𝑏secln
  • E(𝑎𝑥+𝑏)((𝑎𝑥+𝑏))𝑎secln

Q14:

Given that 𝑦=−34(7𝑥+7𝑥)lntansec, determine dd𝑦𝑥.

  • A−347𝑥sec
  • B−347𝑥+47𝑥tansec
  • C−217𝑥+7𝑥7𝑥47𝑥+47𝑥tansecsectansec
  • D−2147𝑥sec

Q15:

Given that 𝑦=8((9𝑥))lnlnln, find dd𝑦𝑥.

  • A−8𝑥9𝑥(9𝑥)lnlnln
  • B−8(9𝑥)lnln
  • C8𝑥9𝑥(9𝑥)lnlnln
  • D8(9𝑥)lnln

Q16:

A factory’s production of 𝑦 units in day 𝑡 is governed by the relation 𝑦=40010−𝑒. What is the rate of change of production with respect to time on the fifth day?

  • A0.8𝑒
  • B320𝑒
  • C−400𝑒
  • D320𝑒

Q17:

Determine the first derivative of 𝑦=−𝑒+2𝑥ln.

  • A−𝑒+2𝑥
  • B−𝑒+16𝑥
  • C−𝑒4+2𝑥
  • D−𝑒4+16𝑥

Q18:

Find the first derivative of the function 𝑦=−5𝑥+2𝑥ln.

  • A−20𝑥+4𝑥(−5𝑥+2𝑥)ln
  • B−20𝑥+4𝑥
  • C20𝑥−4𝑥(5𝑥−2)
  • D𝑥5𝑥−220𝑥−4

Q19:

Find the first derivative of the function 𝑦=−7𝑥6𝑥ln.

  • A−28𝑥6𝑥−76ln
  • B−112𝑥
  • C−28𝑥6𝑥+1ln
  • D−28𝑥6𝑥+1ln

Q20:

Find dd𝑦𝑥, given that 𝑦=4𝑥+34𝑥−7lnln.

  • A−16𝑥(4𝑥−7)ln
  • B−40𝑥(4𝑥−7)ln
  • C−10𝑥(4𝑥−7)ln
  • D−40𝑥(4𝑥−7)ln

Q21:

Differentiate 𝑦=−5|5𝑥−5𝑥+3|ln.

  • A𝑦′=−55𝑥−5𝑥+3
  • B𝑦′=−75𝑥−25𝑥5𝑥−5𝑥+3
  • C𝑦′=−75𝑥−25|5𝑥−5𝑥+3|ln
  • D𝑦′=−25𝑥−25𝑥+1515𝑥−5
  • E𝑦′=−75𝑥−255𝑥−5𝑥+3

Q22:

Differentiate 𝑓(𝑥)=−5√𝑥+4ln, and determine its domain.

  • A𝑓′(𝑥)=−52𝑥√𝑥+4ln, 1𝑒,∞
  • B𝑓′(𝑥)=−5𝑥2√𝑥+4ln, 1𝑒,∞
  • C𝑓′(𝑥)=−5𝑥√𝑥+4ln, 1𝑒,∞
  • D𝑓′(𝑥)=−52𝑥√𝑥+4ln, 1𝑒,∞
  • E𝑓′(𝑥)=−5𝑥2√𝑥+4ln, 1𝑒,∞

Q23:

Differentiate 𝑓(𝑥)=−𝑥+4𝑥ln, and determine its domain.

  • A𝑓′(𝑥)=−1𝑥+4𝑥, ℝ
  • B𝑓′(𝑥)=−2𝑥+4𝑥+4𝑥, ℝ
  • C𝑓′(𝑥)=−1𝑥+4𝑥, (−∞,−4)∪(0,∞)
  • D𝑓′(𝑥)=−𝑥−4𝑥+4𝑥, (−∞,−4)∪(0,∞)
  • E𝑓′(𝑥)=−2𝑥+4𝑥+4𝑥, (−∞,−4)∪(0,∞)

Q24:

Differentiate 𝑓(𝑥)=−4𝑥3(𝑥−3)−3ln, and determine its domain.

  • A𝑓′(𝑥)=−4𝑥(𝑥−3)−12(𝑥−3)+123(𝑥−3)((𝑥−3)−1)lnlnln, (3,𝑒+3)∪(𝑒+3,∞)
  • B𝑓′(𝑥)=−4𝑥(𝑥−3)−8𝑥−12(𝑥−3)+123(𝑥−3)((𝑥−3)−1)lnlnln, (3,𝑒+3)∪(𝑒+3,∞)
  • C𝑓′(𝑥)=4𝑥(𝑥−3)−8𝑥−12(𝑥−3)+12(𝑥−3)((𝑥−3)−1)lnlnln, (3,𝑒+3)∪(𝑒+3,∞)
  • D𝑓′(𝑥)=−4𝑥(𝑥−3)−8𝑥−12(𝑥−3)+123(𝑥−3)((𝑥−3)−1)lnlnln, (−∞,3)∪(3,𝑒+3)∪(𝑒+3,∞)
  • E𝑓′(𝑥)=−4𝑥(𝑥−3)−12(𝑥−3)+123(𝑥−3)((𝑥−3)−1)lnlnln, (−∞,3)∪(3,𝑒+3)∪(𝑒+3,∞)

Q25:

Differentiate 𝑃(𝑣)=2𝑣5𝑣−3ln.

  • A𝑃′(𝑣)=−10𝑣𝑣+10𝑣−6(5𝑣−3)ln
  • B𝑃′(𝑣)=23𝑣
  • C𝑃′(𝑣)=−10𝑣𝑣−10𝑣−6𝑣(5𝑣−3)ln
  • D𝑃′(𝑣)=−10𝑣𝑣+10𝑣−6𝑣(5𝑣−3)ln
  • E𝑃′(𝑣)=−10𝑣𝑣+10𝑣−6𝑣(5𝑣−3)ln

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