Worksheet: Differentiation of Exponential Functions

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

Q1:

Find the first derivative of the function 𝑦 = 4 𝑥 + 2 𝑒 6 𝑥 .

  • A 4 𝑥 + 2 𝑒 5 𝑥
  • B 2 4 𝑥 2 𝑒 5 𝑥
  • C 2 4 𝑥 + 2 5
  • D 2 4 𝑥 + 2 𝑒 5 𝑥

Q2:

Differentiate the function 𝑦 = 3 𝑒 5 𝑥 𝑥 3 .

  • A 𝑦 = 3 𝑒 + 5 3 𝑥 𝑥 2 3
  • B 𝑦 = 3 𝑒 5 3 𝑥 𝑥 4 3
  • C 𝑦 = 3 𝑒 5 3 𝑥 𝑥 2 3
  • D 𝑦 = 3 𝑒 + 5 3 𝑥 𝑥 4 3
  • E 𝑦 = 3 𝑒 + 5 3 𝑥 2 3

Q3:

Find the first derivative of the function 𝑦 = 2 𝑒 5 𝑒 4 𝑥 5 𝑥 4 .

  • A 4 2 𝑒 5 𝑒 4 𝑥 5 𝑥 4
  • B 2 𝑒 5 𝑒 8 𝑒 + 2 5 𝑒 4 𝑥 5 𝑥 3 4 𝑥 5 𝑥
  • C 4 2 𝑒 5 𝑒 8 𝑒 + 2 5 𝑒 4 𝑥 5 𝑥 3 4 𝑥 5 𝑥
  • D 4 2 𝑒 5 𝑒 8 𝑒 + 2 5 𝑒 4 𝑥 5 𝑥 3 4 𝑥 5 𝑥
  • E 4 2 𝑒 5 𝑒 8 𝑒 + 2 5 𝑒 4 𝑥 5 𝑥 4 4 𝑥 5 𝑥

Q4:

Differentiate ( 𝑡 ) = 5 𝑡 𝑒 3 𝑡 .

  • A ( 𝑡 ) = 5 3 𝑡 𝑒 3 𝑡
  • B ( 𝑡 ) = 5 𝑡 𝑒 3 2 𝑡
  • C ( 𝑡 ) = 5 3 𝑡 𝑒 3 4 𝑡
  • D ( 𝑡 ) = 5 3 𝑡 𝑒 3 2 𝑡
  • E ( 𝑡 ) = 5 3 𝑡 𝑒 3 2 𝑡

Q5:

Find d d 𝑦 𝑥 if 5 𝑦 𝑒 = 7 𝑒 2 𝑥 5 .

  • A 𝑦
  • B 2 𝑦
  • C 7 5 𝑦
  • D 2 𝑦
  • E 2 5 𝑦

Q6:

Determine the derivative of 𝑔 ( 𝑥 ) = 5 𝑒 3 𝑥 + 3 𝑥 2 .

  • A 𝑔 ( 𝑥 ) = 5 𝑒 3 𝑥 + 3 𝑥 2
  • B 𝑔 ( 𝑥 ) = 𝑒 1 5 𝑥 1 5 𝑥 3 𝑥 + 3 𝑥 2 2
  • C 𝑔 ( 𝑥 ) = 𝑒 3 0 𝑥 1 5 3 𝑥 + 3 𝑥 2 2
  • D 𝑔 ( 𝑥 ) = 𝑒 ( 3 0 𝑥 1 5 ) 3 𝑥 + 3 𝑥 2
  • E 𝑔 ( 𝑥 ) = 5 𝑒 3 𝑥 + 3 𝑥 2

Q7:

Find the derivative of the function .

  • A
  • B
  • C
  • D
  • E

Q8:

Differentiate 𝑓 ( 𝑥 ) = 𝑒 𝑥 𝑥 s e c .

  • A 𝑓 ( 𝑥 ) = 𝑒 𝑥 + 𝑥 𝑥 2 s e c t a n
  • B 𝑓 ( 𝑥 ) = 𝑒 𝑥 ( 1 𝑥 ) 𝑥 s e c t a n
  • C 𝑓 ( 𝑥 ) = 𝑥 ( 1 + 𝑒 𝑥 ) s e c t a n 𝑥
  • D 𝑓 ( 𝑥 ) = 𝑒 𝑥 ( 1 + 𝑥 ) 𝑥 s e c t a n
  • E 𝑓 ( 𝑥 ) = 𝑒 𝑥 𝑥 𝑥 s e c t a n

Q9:

Find the first derivative of the function 𝑦 = 4 𝑒 7 𝑥 + 4 7 𝑥 .

  • A 2 8 𝑥 𝑒 + 8 4 𝑒 ( 7 𝑥 + 4 ) 7 𝑥 7 𝑥 2
  • B 1 9 6 𝑥 𝑒 + 8 4 𝑒 ( 7 𝑥 + 4 ) 7 𝑥 7 𝑥 2
  • C 1 9 6 𝑥 𝑒 + 8 4 𝑒 7 𝑥 + 4 7 𝑥 7 𝑥
  • D 1 9 6 𝑥 𝑒 + 8 4 𝑒 ( 7 𝑥 + 4 ) 7 𝑥 7 𝑥 2
  • E 1 9 6 𝑒 + 8 4 𝑒 ( 7 𝑥 + 4 ) 7 𝑥 7 𝑥 2

Q10:

Find the first derivative of the function 𝑦 = 𝑒 7 8 𝑥 5 𝑥 t a n .

  • A 𝑒 5 8 𝑥 + 8 8 𝑥 7 8 𝑥 5 𝑥 2 2 t a n s e c t a n
  • B 𝑒 ( 5 8 𝑥 8 8 𝑥 ) 7 8 𝑥 5 𝑥 2 t a n s e c t a n
  • C 𝑒 5 8 𝑥 8 8 𝑥 7 8 𝑥 5 𝑥 2 t a n s e c t a n
  • D 𝑒 5 8 𝑥 8 8 𝑥 7 8 𝑥 5 𝑥 2 2 t a n s e c t a n
  • E 𝑒 8 𝑥 8 8 𝑥 7 8 𝑥 5 𝑥 2 2 t a n s e c t a n

Q11:

Find the derivative of the function 𝑓 ( 𝑧 ) = 3 𝑒 4 𝑧 4 𝑧 + 1 .

  • A 𝑓 ( 𝑧 ) = 3 𝑒 ( 4 𝑧 + 1 ) 4 𝑧 4 𝑧 + 1 2
  • B 𝑓 ( 𝑧 ) = 1 2 𝑒 4 𝑧 + 1 4 𝑧 4 𝑧 + 1
  • C 𝑓 ( 𝑧 ) = 1 2 𝑒 ( 4 𝑧 + 1 ) 4 𝑧 4 𝑧 + 1 2
  • D 𝑓 ( 𝑧 ) = 1 2 𝑒 ( 4 𝑧 + 1 ) 4 𝑧 4 𝑧 + 1 2
  • E 𝑓 ( 𝑧 ) = 3 𝑒 ( 4 𝑧 + 1 ) 4 𝑧 4 𝑧 + 1 2

Q12:

Find the derivative of the function 𝑦 = 5 𝑒 2 𝜃 t a n .

  • A 𝑦 = 1 0 𝑒 𝜃 2 𝜃 2 t a n c s c
  • B 𝑦 = 1 0 𝑒 𝜃 2 𝜃 2 t a n s e c
  • C 𝑦 = 1 0 𝑒 𝜃 2 𝜃 2 t a n c s c
  • D 𝑦 = 1 0 𝑒 𝜃 2 𝜃 2 t a n s e c
  • E 𝑦 = 1 0 𝑒 𝜃 2 𝜃 t a n t a n

Q13:

Find the first derivative of the function 𝑦 = 3 𝑒 4 7 𝑥 s e c .

  • A 8 4 𝑒 7 𝑥 4 7 𝑥 2 s e c t a n
  • B 3 𝑒 4 7 𝑥 s e c
  • C 8 4 𝑒 7 𝑥 7 𝑥 4 7 𝑥 s e c s e c t a n
  • D 8 4 𝑒 7 𝑥 7 𝑥 4 7 𝑥 s e c s e c t a n
  • E 1 2 𝑒 7 𝑥 7 𝑥 4 7 𝑥 s e c s e c t a n

Q14:

Find the derivative of the function .

  • A
  • B
  • C
  • D
  • E

Q15:

Find the derivative of the function 𝑦 = 4 𝑒 + 5 𝑒 + 5 c o s 𝑥 𝑥 .

  • A 𝑦 = 4 0 𝑒 𝑒 + 5 𝑒 + 5 𝑒 + 5 𝑥 𝑥 𝑥 𝑥 s i n
  • B 𝑦 = 4 0 𝑒 ( 𝑒 + 5 ) 𝑒 + 5 𝑒 + 5 𝑥 𝑥 2 𝑥 𝑥 s i n
  • C 𝑦 = 4 𝑒 + 5 𝑒 + 5 s i n 𝑥 𝑥
  • D 𝑦 = 4 0 𝑒 ( 𝑒 + 5 ) 𝑒 + 5 𝑒 + 5 𝑥 𝑥 2 𝑥 𝑥 s i n
  • E 𝑦 = 4 𝑒 + 5 𝑒 + 5 s i n 𝑥 𝑥

Q16:

Find the derivative of the function 𝑓 ( 𝑡 ) = 𝑒 s i n 2 5 𝑡 s i n 2 .

  • A 𝑓 ( 𝑡 ) = 1 0 𝑒 𝑒 5 𝑡 𝑒 5 𝑡 s i n s i n s i n 2 2 2 5 𝑡 5 𝑡 5 𝑡 s i n s i n c o s c o s
  • B 𝑓 ( 𝑡 ) = 2 0 𝑒 𝑒 5 𝑡 𝑒 5 𝑡 s i n s i n s i n 2 2 2 5 𝑡 5 𝑡 5 𝑡 s i n s i n c o s c o s
  • C 𝑓 ( 𝑡 ) = 1 0 𝑒 𝑒 5 𝑡 𝑒 5 𝑡 s i n s i n s i n 2 2 2 5 𝑡 5 𝑡 5 𝑡 s i n s i n c o s c o s
  • D 𝑓 ( 𝑡 ) = 2 0 𝑒 𝑒 5 𝑡 𝑒 5 𝑡 s i n s i n s i n 2 2 2 5 𝑡 5 𝑡 5 𝑡 s i n s i n c o s c o s
  • E 𝑓 ( 𝑡 ) = 2 𝑒 𝑒 𝑒 s i n s i n s i n 2 2 2 5 𝑡 5 𝑡 5 𝑡 s i n c o s

Q17:

Find 𝑓 ( 4 ) , if 𝑓 ( 𝑥 ) = 8 𝑒 6 𝑥 .

  • A 1 4 𝑓 ( 4 )
  • B 8 𝑓 ( 4 )
  • C 4 8 𝑓 ( 4 )
  • D 6 𝑓 ( 4 )

Q18:

If 𝑓 ( 𝑥 ) = 5 𝑒 9 𝑥 , find 𝑓 ( 𝑥 ) .

  • A 1 8 𝑒 9 𝑥
  • B 𝑒 9 𝑥
  • C 5 𝑒 1 0 𝑥
  • D 4 5 𝑒 9 𝑥
  • E 5 𝑒 8 𝑥

Q19:

Differentiate 𝑧 ( 𝑦 ) = 𝐴 5 𝑦 + 𝐵 𝑒 3 3 𝑦 .

  • A d d 𝑧 𝑦 = 𝐴 5 𝑦 + 𝐵 𝑒 4 3 𝑦
  • B d d 𝑧 𝑦 = 3 𝐴 5 𝑦 + 3 𝐵 𝑒 2 3 𝑦
  • C d d 𝑧 𝑦 = 3 𝐴 5 𝑦 + 3 𝐵 𝑒 3 3 𝑦
  • D d d 𝑧 𝑦 = 3 𝐴 5 𝑦 + 3 𝐵 𝑒 4 3 𝑦
  • E d d 𝑧 𝑦 = 𝐴 5 𝑦 + 𝐵 𝑒 2 3 𝑦

Q20:

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

  • A 4 𝑥 𝑒 3 ( 4 𝑥 3 ) 1 3 3
  • B 𝑒 1 3 3 ( 4 𝑥 3 )
  • C 3 6 𝑥 𝑒 2 ( 4 𝑥 3 ) 1 3 3
  • D 4 𝑥 𝑒 2 ( 4 𝑥 3 ) 1 3 3

Q21:

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

Q22:

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

Q23:

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

Q24:

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

Q25:

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

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