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Worksheet: Derivatives of Logarithmic Functions

Q1:

Find d d 𝑦 π‘₯ , given that 𝑦 = 6 6 π‘₯ l o g 6 .

  • A 6 6 l n
  • B 6 π‘₯ 6 l o g
  • C 6 6 l o g
  • D 6 π‘₯ 6 l n

Q2:

Find d d 𝑦 π‘₯ , given that 𝑦 = βˆ’ 8 π‘₯ l o g 8 .

  • A 8 8 l n
  • B βˆ’ 1 π‘₯ 8 l o g
  • C 8 8 l o g
  • D βˆ’ 1 π‘₯ 8 l n

Q3:

Find d d 𝑦 π‘₯ , given that 𝑦 = ( π‘₯ + 7 ) l n 2 .

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

Q4:

Let 𝐴 and 𝐡 , respectively, be the 𝑦 - and π‘₯ -intercepts of the tangent line to 𝑦 = 2 5 π‘₯ l n at π‘₯ = 2 . What is the length of the segment 𝐴 𝐡 ?

  • A 2 √ 2 ( βˆ’ 2 + 2 1 0 ) l n length units
  • B 2 ( βˆ’ 2 1 0 + 2 ) l n 2 length units
  • C 4 ( βˆ’ 2 1 0 + 2 ) l n 2 length units
  • D √ 2 ( βˆ’ 2 + 2 1 0 ) l n length units

Q5:

Differentiate 𝑓 ( π‘₯ ) = βˆ’ 3 2 π‘₯ l n .

  • A 𝑓 β€² ( π‘₯ ) = 3 2 π‘₯ l n 2
  • B 𝑓 β€² ( π‘₯ ) = βˆ’ 3 2 π‘₯ π‘₯ l n 2
  • C 𝑓 β€² ( π‘₯ ) = 3 2 π‘₯ π‘₯ l n
  • D 𝑓 β€² ( π‘₯ ) = 3 2 π‘₯ π‘₯ l n 2
  • E 𝑓 β€² ( π‘₯ ) = βˆ’ 3 2 π‘₯ π‘₯ l n

Q6:

Differentiate 𝑓 ( π‘₯ ) = 5 2 π‘₯ l n .

  • A 𝑓 β€² ( π‘₯ ) = βˆ’ 5 2 π‘₯ l n 2
  • B 𝑓 β€² ( π‘₯ ) = 5 2 π‘₯ π‘₯ l n 2
  • C 𝑓 β€² ( π‘₯ ) = βˆ’ 5 2 π‘₯ π‘₯ l n
  • D 𝑓 β€² ( π‘₯ ) = βˆ’ 5 2 π‘₯ π‘₯ l n 2
  • E 𝑓 β€² ( π‘₯ ) = 5 2 π‘₯ π‘₯ l n

Q7:

Find d d 𝑦 π‘₯ , given that π‘₯ 4 𝑦 = 1 2 5 l n .

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

Q8:

Determine the equation of the tangent to the curve 𝑦 = π‘₯ + 4 π‘₯ 3 l n at π‘₯ = 1 .

  • A π‘₯ 7 + 𝑦 βˆ’ 8 7 = 0
  • B 7 π‘₯ + 𝑦 βˆ’ 8 = 0
  • C βˆ’ π‘₯ 7 + 𝑦 βˆ’ 6 7 = 0
  • D βˆ’ 7 π‘₯ + 𝑦 + 6 = 0

Q9:

Determine the equation of the tangent to the curve 𝑦 = 4 π‘₯ βˆ’ 7 π‘₯ 5 l n at π‘₯ = 1 .

  • A π‘₯ 1 3 + 𝑦 βˆ’ 5 3 1 3 = 0
  • B 1 3 π‘₯ + 𝑦 βˆ’ 1 7 = 0
  • C βˆ’ π‘₯ 1 3 + 𝑦 βˆ’ 5 1 1 3 = 0
  • D βˆ’ 1 3 π‘₯ + 𝑦 + 9 = 0

Q10:

Use logarithmic differentiation to find the derivative of the function 𝑦 = 4 𝑒 π‘₯ 3 π‘₯ + 5 π‘₯ + 3 π‘₯ 2 2 c o s .

  • A 𝑦 β€² = 4 𝑒 π‘₯ 3 π‘₯ + 5 π‘₯ + 3 ο€Ό 6 π‘₯ + 5 3 π‘₯ + 5 π‘₯ + 3 βˆ’ 2 π‘₯ + 1  π‘₯ 2 2 2 c o s t a n
  • B 𝑦 β€² = 4 𝑒 π‘₯ 3 π‘₯ + 5 π‘₯ + 3 ο€Ό βˆ’ 6 π‘₯ 3 π‘₯ + 5 π‘₯ + 3 βˆ’ 2 π‘₯  π‘₯ 2 2 2 c o s t a n
  • C 𝑦 β€² = 4 𝑒 π‘₯ 3 π‘₯ + 5 π‘₯ + 3 ο€Ό 6 π‘₯ + 5 3 π‘₯ + 5 π‘₯ + 3 βˆ’ 2 π‘₯  π‘₯ 2 2 2 c o s t a n
  • D 𝑦 β€² = 4 𝑒 π‘₯ 3 π‘₯ + 5 π‘₯ + 3 ο€Ό βˆ’ 6 π‘₯ + 5 3 π‘₯ + 5 π‘₯ + 3 βˆ’ 2 π‘₯ + 1  π‘₯ 2 2 2 c o s t a n
  • E 𝑦 β€² = 4 𝑒 π‘₯ 3 π‘₯ + 5 π‘₯ + 3 ο€Ό βˆ’ 6 π‘₯ 3 π‘₯ + 5 π‘₯ + 3 + 2 π‘₯ + 1  π‘₯ 2 2 2 c o s t a n

Q11:

Use logarithmic differentiation to find the derivative of the function 𝑦 = 5 𝑒 π‘₯ 2 π‘₯ + 2 π‘₯ + 3 5 π‘₯ 2 2 c o s .

  • A 𝑦 β€² = 5 𝑒 π‘₯ 2 π‘₯ + 2 π‘₯ + 3 ο€Ό 4 π‘₯ + 2 2 π‘₯ + 2 π‘₯ + 3 βˆ’ 2 π‘₯ + 5  5 π‘₯ 2 2 2 c o s t a n
  • B 𝑦 β€² = 5 𝑒 π‘₯ 2 π‘₯ + 2 π‘₯ + 3 ο€Ό βˆ’ 4 π‘₯ 2 π‘₯ + 2 π‘₯ + 3 βˆ’ 2 π‘₯  5 π‘₯ 2 2 2 c o s t a n
  • C 𝑦 β€² = 5 𝑒 π‘₯ 2 π‘₯ + 2 π‘₯ + 3 ο€Ό 4 π‘₯ + 2 2 π‘₯ + 2 π‘₯ + 3 βˆ’ 2 π‘₯  5 π‘₯ 2 2 2 c o s t a n
  • D 𝑦 β€² = 5 𝑒 π‘₯ 2 π‘₯ + 2 π‘₯ + 3 ο€Ό βˆ’ 4 π‘₯ + 2 2 π‘₯ + 2 π‘₯ + 3 βˆ’ 2 π‘₯ + 5  5 π‘₯ 2 2 2 c o s t a n
  • E 𝑦 β€² = 5 𝑒 π‘₯ 2 π‘₯ + 2 π‘₯ + 3 ο€Ό βˆ’ 4 π‘₯ 2 π‘₯ + 2 π‘₯ + 3 + 2 π‘₯ + 5  5 π‘₯ 2 2 2 c o s t a n

Q12:

Differentiate the function 𝐹 ( 𝑠 ) = 𝑠 l n l n .

  • A 𝐹 β€² ( 𝑠 ) = βˆ’ 1 𝑠 𝑠 l n
  • B 𝐹 β€² ( 𝑠 ) = 1 𝑠 l n
  • C 𝐹 β€² ( 𝑠 ) = 1 𝑠 l n l n
  • D 𝐹 β€² ( 𝑠 ) = 1 𝑠 𝑠 l n
  • E 𝐹 β€² ( 𝑠 ) = 𝑠 𝑠 l n

Q13:

Determine the equation of the tangent to the curve 𝑦 = βˆ’ 2 ο€» √ 2 π‘₯ + 2  l n c o s at π‘₯ = 3 πœ‹ 4 .

  • A 2 π‘₯ + 𝑦 βˆ’ 3 πœ‹ 2 = 0
  • B 2 π‘₯ + 𝑦 + 3 πœ‹ 2 = 0
  • C βˆ’ 2 π‘₯ + 𝑦 βˆ’ 3 πœ‹ 2 = 0
  • D βˆ’ 2 π‘₯ + 𝑦 + 3 πœ‹ 2 = 0

Q14:

Determine the equation of the tangent to the curve 𝑦 = βˆ’ 2 ο€» βˆ’ √ 2 π‘₯ + 2  l n c o s at π‘₯ = πœ‹ 4 .

  • A βˆ’ 2 π‘₯ + 𝑦 + πœ‹ 2 = 0
  • B βˆ’ 2 π‘₯ + 𝑦 βˆ’ πœ‹ 2 = 0
  • C 2 π‘₯ + 𝑦 + πœ‹ 2 = 0
  • D 2 π‘₯ + 𝑦 βˆ’ πœ‹ 2 = 0

Q15:

Use logarithmic differentiation to find the derivative of the function 𝑦 = βˆ’ 4 π‘₯ 5 π‘₯ s i n .

  • A 𝑦 β€² = βˆ’ 2 0 π‘₯  π‘₯ π‘₯ βˆ’ π‘₯ π‘₯  5 π‘₯ s i n s i n c o s l n
  • B 𝑦 β€² = π‘₯ π‘₯ + π‘₯ π‘₯ s i n c o s l n
  • C 𝑦 β€² = 2 0 π‘₯  π‘₯ π‘₯ + π‘₯ π‘₯  5 π‘₯ s i n s i n c o s l n
  • D 𝑦 β€² = βˆ’ 2 0 π‘₯  π‘₯ π‘₯ + π‘₯ π‘₯  5 π‘₯ s i n s i n c o s l n
  • E 𝑦 β€² = 2 0 π‘₯  π‘₯ π‘₯ βˆ’ π‘₯ π‘₯  5 π‘₯ s i n s i n c o s l n

Q16:

Use logarithmic differentiation to find the derivative of the function 𝑦 = 2 π‘₯ 4 π‘₯ s i n .

  • A 𝑦 β€² = 8 π‘₯  π‘₯ π‘₯ βˆ’ π‘₯ π‘₯  4 π‘₯ s i n s i n c o s l n
  • B 𝑦 β€² = π‘₯ π‘₯ + π‘₯ π‘₯ s i n c o s l n
  • C 𝑦 β€² = βˆ’ 8 π‘₯  π‘₯ π‘₯ + π‘₯ π‘₯  4 π‘₯ s i n s i n c o s l n
  • D 𝑦 β€² = 8 π‘₯  π‘₯ π‘₯ + π‘₯ π‘₯  4 π‘₯ s i n s i n c o s l n
  • E 𝑦 β€² = βˆ’ 8 π‘₯  π‘₯ π‘₯ βˆ’ π‘₯ π‘₯  4 π‘₯ s i n s i n c o s l n

Q17:

Use logarithmic differentiation to determine the derivative of the function 𝑦 = βˆ’ 3 π‘₯ 2 π‘₯ .

  • A 2 π‘₯ π‘₯ l n
  • B βˆ’ 3 π‘₯ 2 π‘₯ + 2 2 π‘₯ l n
  • C 2 ( π‘₯ + 1 ) l n
  • D βˆ’ 6 π‘₯ ( π‘₯ + 1 ) 2 π‘₯ l n
  • E βˆ’ 6 π‘₯ 2 π‘₯

Q18:

Use logarithmic differentiation to determine the derivative of the function 𝑦 = βˆ’ π‘₯ π‘₯ .

  • A π‘₯ π‘₯ l n
  • B βˆ’ π‘₯ π‘₯ + 1 π‘₯ l n
  • C l n π‘₯ + 1
  • D βˆ’ π‘₯ ( π‘₯ + 1 ) π‘₯ l n
  • E βˆ’ π‘₯ π‘₯

Q19:

Use logarithmic differentiation to find the derivative of the function 𝑦 = 5 ( π‘₯ ) s i n 3 π‘₯ l n .

  • A 𝑦 β€² = 1 5 ( π‘₯ )  π‘₯ π‘₯ + π‘₯ π‘₯  s i n t a n l n l n s i n 3 π‘₯ l n
  • B 𝑦 β€² = 1 5 ( π‘₯ )  π‘₯ π‘₯ + π‘₯ π‘₯  s i n l n s i n l n s i n 3 π‘₯ l n
  • C 𝑦 β€² = 1 5 ( π‘₯ )  π‘₯ π‘₯ βˆ’ π‘₯ π‘₯  s i n l n s i n c o t l n 3 π‘₯ l n
  • D 𝑦 β€² = 1 5 ( π‘₯ )  π‘₯ π‘₯ + π‘₯ π‘₯  s i n c o t l n l n s i n 3 π‘₯ l n
  • E 𝑦 β€² = 3  π‘₯ π‘₯ + π‘₯ π‘₯  c o t l n l n s i n

Q20:

Use logarithmic differentiation to find the derivative of the function 𝑦 = βˆ’ 2 ( π‘₯ ) s i n 5 π‘₯ l n .

  • A 𝑦 β€² = βˆ’ 1 0 ( π‘₯ )  π‘₯ π‘₯ + π‘₯ π‘₯  s i n t a n l n l n s i n 5 π‘₯ l n
  • B 𝑦 β€² = βˆ’ 1 0 ( π‘₯ )  π‘₯ π‘₯ + π‘₯ π‘₯  s i n l n s i n l n s i n 5 π‘₯ l n
  • C 𝑦 β€² = βˆ’ 1 0 ( π‘₯ )  π‘₯ π‘₯ βˆ’ π‘₯ π‘₯  s i n l n s i n c o t l n 5 π‘₯ l n
  • D 𝑦 β€² = βˆ’ 1 0 ( π‘₯ )  π‘₯ π‘₯ + π‘₯ π‘₯  s i n c o t l n l n s i n 5 π‘₯ l n
  • E 𝑦 β€² = 5  π‘₯ π‘₯ + π‘₯ π‘₯  c o t l n l n s i n

Q21:

Find d d 3 3 𝑦 π‘₯ , given that 𝑦 = 5 8 4 π‘₯ l n .

  • A 5 8 π‘₯ 3
  • B 5 π‘₯ 4 3
  • C βˆ’ 5 4 π‘₯ 3
  • D 5 4 π‘₯ 3

Q22:

Find d d 3 3 𝑦 π‘₯ , given that 𝑦 = 1 2 6 π‘₯ l n .

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

Q23:

If 𝑦 = βˆ’ 2 π‘₯ + 1 6 4 π‘₯ 2 l n , find the value of π‘₯ at which the tangent to the curve is parallel to the π‘₯ -axis.

Q24:

If 𝑦 = βˆ’ π‘₯ + 2 4 3 π‘₯ 3 l n , find the value of π‘₯ at which the tangent to the curve is parallel to the π‘₯ -axis.

Q25:

Use logarithmic differentiation to find the derivative of 𝑦 , given that 𝑦 = √ 5 π‘₯ 𝑒 ( π‘₯ + 5 ) π‘₯ + 2 π‘₯ 2 2 3 .

  • A 𝑦 β€² = √ 5 π‘₯ 𝑒 ( π‘₯ + 5 ) ο€Ό π‘₯ + 2 + 1 2 π‘₯ + 2 3 π‘₯ + 1 5  π‘₯ + 2 π‘₯ 2 2 3
  • B 𝑦 β€² = √ 5 π‘₯ 𝑒 ( π‘₯ + 5 ) ο€Ό 2 π‘₯ + 2 + 5 2 π‘₯ + 2 3 π‘₯ + 1 5  π‘₯ + 2 π‘₯ 2 2 3
  • C 𝑦 β€² = √ 5 π‘₯ 𝑒 ( π‘₯ + 5 ) ο€Ό π‘₯ + 2 + 5 2 π‘₯ + 2 3 π‘₯ + 1 5  π‘₯ + 2 π‘₯ 2 2 3
  • D 𝑦 β€² = √ 5 π‘₯ 𝑒 ( π‘₯ + 5 ) ο€Ό 2 π‘₯ + 2 + 1 2 π‘₯ + 2 3 π‘₯ + 1 5  π‘₯ + 2 π‘₯ 2 2 3
  • E 𝑦 β€² = √ 5 π‘₯ 𝑒 ( π‘₯ + 5 ) ο€Ό 2 π‘₯ + 2 + 1 2 π‘₯ + 2 3 π‘₯ + 1 5  π‘₯ + 2 π‘₯ 2 2 2 3