Worksheet: Differentiation of General Exponential Functions

In this worksheet, we will practice differentiating general exponential functions without taking limits of the function.

Q1:

If 𝑦 = 3 × 2 𝑥 , determine d d 𝑦 𝑥 .

  • A 6 2 l n 𝑥
  • B 3 × 2 2 𝑥 𝑥 l n
  • C 6 2 l n
  • D 3 × 2 2 𝑥 l n

Q2:

Find d d 𝑦 𝑥 if 𝑦 = 4 ( 9 𝑥 4 𝑥 + 8 ) 2 .

  • A 4 4 l n ( 9 𝑥 4 𝑥 + 8 ) 2
  • B 4 4 ( 9 𝑥 4 𝑥 + 8 ) 2 l n
  • C 8 ( 9 𝑥 2 ) 4 l n ( 9 𝑥 4 𝑥 + 8 ) 2
  • D 2 ( 9 𝑥 2 ) 4 4 ( 9 𝑥 4 𝑥 + 8 ) 2 l n

Q3:

Find the derivative of the function 𝐺 ( 𝑥 ) = 2 𝐶 𝑥 .

  • A 𝐺 ( 𝑥 ) = 𝐶 2 𝑥 𝐶 𝑥 2
  • B 𝐺 ( 𝑥 ) = 𝐶 ( 2 ) 2 𝑥 l n 𝐶 𝑥 2
  • C 𝐺 ( 𝑥 ) = 𝐶 ( 2 ) 2 𝑥 l n 𝐶 𝑥
  • D 𝐺 ( 𝑥 ) = 𝐶 ( 2 ) 2 𝑥 l n 𝐶 𝑥 2
  • E 𝐺 ( 𝑥 ) = 𝐶 ( 2 ) 2 𝑥 l n 𝐶 𝑥

Q4:

Determine the derivative of 𝑟 ( 𝑡 ) = 8 6 𝑡 .

  • A 𝑟 ( 𝑡 ) = 8 8 6 𝑡 l n
  • B 𝑟 ( 𝑡 ) = 6 × 8 𝑡 6 𝑡 1
  • C 𝑟 ( 𝑡 ) = 8 8 2 𝑡 6 𝑡 l n
  • D 𝑟 ( 𝑡 ) = 3 × 8 8 𝑡 6 𝑡 l n
  • E 𝑟 ( 𝑡 ) = 3 6 × 8 6 𝑡

Q5:

Find d d 𝑦 𝑥 if 𝑦 = 4 8 𝑥 4 8 𝑥 8 4 .

  • A 4 8 𝑥 4 4 + 6 4 𝑥 8 𝑥 8 7 4 l n
  • B 4 3 2 𝑥 8 𝑥 4 + 6 4 𝑥 8 𝑥 3 8 7 4
  • C 4 3 2 8 𝑥 4 4 + 6 4 𝑥 8 𝑥 8 7 4 l n
  • D 4 3 2 𝑥 8 𝑥 4 4 + 6 4 𝑥 8 𝑥 3 8 7 4 l n
  • E 3 2 𝑥 8 𝑥 4 4 + 6 4 𝑥 3 8 7 l n

Q6:

Given 𝑦 = 9 𝑥 + 6 𝑥 𝑥 4 , find d d 𝑦 𝑥 .

  • A 9 2 4 𝑥 + 1 2 𝑥 + 𝑥 + 6 𝑥 9 𝑥 3 4 l n
  • B 9 𝑥 + 6 𝑥 9 + 9 6 𝑥 + 1 𝑥 𝑥 4 𝑥 3 l n
  • C 9 𝑥 + 6 𝑥 + 9 2 4 𝑥 + 1 2 𝑥 𝑥 4 𝑥 3
  • D 9 𝑥 + 6 𝑥 9 + 9 2 4 𝑥 + 1 2 𝑥 𝑥 4 𝑥 3 l n

Q7:

Find d d 𝑦 𝑥 if 𝑦 = 3 6 s e c 2 𝑥 .

  • A 7 2 × 3 6 𝑥 𝑥 l n t a n s e c s e c 2 𝑥 2
  • B 3 6 × 3 6 𝑥 𝑥 s e c 2 𝑥 2 l n t a n s e c
  • C 3 6 × 2 3 6 𝑥 𝑥 s e c 2 𝑥 l n t a n s e c
  • D 3 6 × 2 3 6 𝑥 𝑥 s e c 2 𝑥 2 l n t a n s e c

Q8:

Find the first derivative of the function 𝑦 = 7 9 𝑥 8 2 .

  • A 2 7 9 𝑥 8 3
  • B 7 7 1 8 𝑥 + 1 6 l n
  • C 2 7 l n 9 𝑥 8
  • D 1 8 7 7 1 8 𝑥 + 1 6 l n

Q9:

Given that 𝑦 = 9 5 𝑥 2 𝑒 s i n 2 𝑥 , find d d 𝑦 𝑥 .

  • A 9 2 𝑒 ( 2 5 𝑥 + 5 𝑥 ) 2 𝑥 s i n c o s
  • B 9 2 𝑒 ( 2 5 𝑥 + 5 5 𝑥 ) 2 𝑥 s i n c o s
  • C 9 2 𝑒 ( 2 5 𝑥 + 5 𝑥 ) 2 𝑥 s i n c o s
  • D 9 2 𝑒 ( 2 5 𝑥 + 5 5 𝑥 ) 2 𝑥 s i n c o s

Q10:

Find the derivative of the function 𝑦 = 2 5 7 𝑥 .

  • A 𝑦 = 2 5 7 2 l n l n l n 5 7 𝑥
  • B 𝑦 = 7 2 5 7 l n 5 7 𝑥 7 𝑥 𝑥
  • C 𝑦 = 5 2 2 7 5 𝑥 7 𝑥 l n
  • D 𝑦 = 2 5 7 2 5 7 l n l n l n 5 7 𝑥 7 𝑥 𝑥
  • E 𝑦 = 2 2 5 7 𝑥 l n

Q11:

Determine the derivative of 𝑓 ( 𝑡 ) = 6 𝑡 7 .

  • A 𝑓 ( 𝑡 ) = 6 6 𝑡 7 l n
  • B 𝑓 ( 𝑡 ) = 6 𝑡 6 𝑡 1 7 7 l n
  • C 𝑓 ( 𝑡 ) = 6 𝑡 6 𝑡 6 7 l n
  • D 𝑓 ( 𝑡 ) = 7 6 𝑡 6 𝑡 6 7 l n
  • E 𝑓 ( 𝑡 ) = 7 6 𝑡 𝑡 6 7

Q12:

Find the derivative of the function 𝑔 ( 𝑥 ) = ( 2 𝑟 𝑎 + 𝑛 ) 𝑟 𝑥 𝑝 , where 𝑎 , 𝑝 , 𝑟 , and 𝑛 are constants.

  • A 𝑔 ( 𝑥 ) = 2 𝑟 𝑝 ( 2 𝑟 𝑎 + 𝑛 ) 𝑎 𝑟 𝑥 𝑝 1 𝑟 𝑥
  • B 𝑔 ( 𝑥 ) = 2 𝑟 𝑝 ( 𝑎 ) ( 2 𝑟 𝑎 + 𝑛 ) 𝑎 l n 𝑟 𝑥 𝑝 1 𝑟 𝑥
  • C 𝑔 ( 𝑥 ) = 2 𝑟 𝑝 ( 𝑎 ) ( 2 𝑟 𝑎 + 𝑛 ) 2 𝑟 𝑥 𝑝 l n
  • D 𝑔 ( 𝑥 ) = 2 𝑟 𝑝 ( 𝑎 ) ( 2 𝑟 𝑎 + 𝑛 ) 𝑎 2 𝑟 𝑥 𝑝 1 𝑟 𝑥 l n
  • E 𝑔 ( 𝑥 ) = 2 𝑟 𝑝 ( 2 𝑟 𝑎 + 𝑛 ) 2 𝑟 𝑥 𝑝 1

Q13:

Find d d 𝑦 𝑥 if 𝑦 = 8 ( 3 𝑥 5 𝑥 + 8 ) 2 .

  • A 8 8 l n ( 3 𝑥 5 𝑥 + 8 ) 2
  • B 8 8 ( 3 𝑥 5 𝑥 + 8 ) 2 l n
  • C 8 ( 6 𝑥 5 ) 8 l n ( 3 𝑥 5 𝑥 + 8 ) 2
  • D ( 6 𝑥 5 ) 8 8 ( 3 𝑥 5 𝑥 + 8 ) 2 l n

Q14:

Find the derivative of the function 𝑦 = 3 2 7 𝑥 .

  • A 𝑦 = 2 3 7 3 l n l n l n 2 7 𝑥
  • B 𝑦 = 7 3 2 7 l n 2 7 𝑥 7 𝑥 𝑥
  • C 𝑦 = 2 3 3 7 2 𝑥 7 𝑥 l n
  • D 𝑦 = 2 3 7 3 2 7 l n l n l n 2 7 𝑥 7 𝑥 𝑥
  • E 𝑦 = 3 3 2 7 𝑥 l n

Q15:

Determine the derivative of 𝑟 ( 𝑡 ) = 9 7 𝑡 .

  • A 𝑟 ( 𝑡 ) = 9 9 7 𝑡 l n
  • B 𝑟 ( 𝑡 ) = 7 × 9 𝑡 7 𝑡 1
  • C 𝑟 ( 𝑡 ) = 9 9 2 𝑡 7 𝑡 l n
  • D 𝑟 ( 𝑡 ) = 7 × 9 9 2 𝑡 7 𝑡 l n
  • E 𝑟 ( 𝑡 ) = 4 9 × 9 7 𝑡

Q16:

Given 𝑦 = 5 5 𝑥 + 9 𝑥 𝑥 3 , find d d 𝑦 𝑥 .

  • A 5 2 7 𝑥 + 5 2 𝑥 + 5 𝑥 + 9 𝑥 5 𝑥 2 3 l n
  • B 5 5 𝑥 + 9 𝑥 5 + 5 9 𝑥 + 5 𝑥 𝑥 3 𝑥 2 l n
  • C 5 5 𝑥 + 9 𝑥 + 5 2 7 𝑥 + 5 2 𝑥 𝑥 3 𝑥 2
  • D 5 5 𝑥 + 9 𝑥 5 + 5 2 7 𝑥 + 5 2 𝑥 𝑥 3 𝑥 2 l n

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