Worksheet: Second- and Higher-Order Derivatives

In this worksheet, we will practice finding second- and higher-order derivatives of a function including using differentiation rules.

Q1:

Find the first and second derivatives of the function 𝐺 ( 𝑟 ) = 3 𝑟 5 𝑟 .

  • A 𝐺 ( 𝑟 ) = 3 𝑟 5 𝑟 , 𝐺 ( 𝑟 ) = 3 𝑟 5 𝑟
  • B 𝐺 ( 𝑟 ) = 3 𝑟 5 𝑟 , 𝐺 ( 𝑟 ) = 3 2 𝑟 + 4 𝑟
  • C 𝐺 ( 𝑟 ) = 3 2 𝑟 𝑟 , 𝐺 ( 𝑟 ) = 3 4 𝑟 + 4 5 𝑟
  • D 𝐺 ( 𝑟 ) = 3 2 𝑟 𝑟 , 𝐺 ( 𝑟 ) = 3 4 𝑟 + 4 5 𝑟
  • E 𝐺 ( 𝑟 ) = 3 2 𝑟 𝑟 , 𝐺 ( 𝑟 ) = 3 4 𝑟 + 4 5 𝑟

Q2:

Given that 𝑦 = 𝑎 𝑥 + 𝑏 𝑥 , 𝑦 = 1 8 , and 𝑦 𝑥 = 1 4 d d , find 𝑎 and 𝑏 .

  • A 𝑎 = 6 , 𝑏 = 2 9
  • B 𝑎 = 3 , 𝑏 = 2 5
  • C 𝑎 = 6 , 𝑏 = 4 3
  • D 𝑎 = 3 , 𝑏 = 1 1

Q3:

Find the third derivative of the function 𝑦 = 4 4 𝑥 2 𝑥 s i n .

  • A 1 7 6 𝑥 2 𝑥 + 1 7 6 2 𝑥 s i n c o s
  • B 3 5 2 𝑥 2 𝑥 + 5 2 8 2 𝑥 c o s s i n
  • C 1 7 6 𝑥 2 𝑥 1 7 6 2 𝑥 s i n c o s
  • D 3 5 2 𝑥 2 𝑥 5 2 8 2 𝑥 c o s s i n
  • E 8 𝑥 2 𝑥 c o s

Q4:

Given 𝑦 = 𝑥 9 , find d d 𝑦 𝑥 .

  • A 1 2 ( 𝑥 9 )
  • B 3 4 ( 𝑥 9 )
  • C 4 ( 𝑥 9 )
  • D 1 4 ( 𝑥 9 )

Q5:

Given that 𝑦 = ( 𝑥 7 ) ( 4 𝑥 + 7 ) , and 𝑧 = 𝑥 + 5 𝑥 + 9 , determine d d d d 𝑦 𝑥 + 𝑧 𝑥 .

Q6:

If 𝑓 ( 𝑥 ) = 𝑎 𝑥 + 7 𝑥 8 𝑥 + 9 , and 𝑓 ( 9 ) = 9 , find 𝑎 .

  • A 8 9
  • B 1 6
  • C 1 6 9
  • D 2 3 5 4

Q7:

Determine the value of the second derivative of the function 𝑦 = 1 2 𝑥 8 𝑥 at ( 1 , 4 ) .

  • A16
  • B48
  • C 8
  • D 1 6

Q8:

If 𝑦 = 5 𝑥 s i n , find 2 5 𝑦 𝑥 + 𝑦 𝑥 d d d d .

Q9:

Given that 𝑦 = 3 𝑥 5 2 𝑥 + 7 , determine d d 𝑦 𝑥 .

  • A 6 2 𝑥 ( 2 𝑥 + 7 )
  • B 6 2 7 6 𝑥 ( 2 𝑥 + 7 )
  • C 7 6 𝑥 ( 2 𝑥 + 7 )
  • D 6 2 7 6 𝑥 ( 2 𝑥 + 7 )
  • E 6 2 7 + 6 𝑥 ( 2 𝑥 + 7 )

Q10:

Find the third derivative of the function 𝑦 = 1 1 𝑥 + 1 4 𝑥 .

  • A 1 4 𝑥
  • B 2 8 𝑥
  • C 8 4 𝑥
  • D 8 4 𝑥

Q11:

Find the first and second derivatives of the function 𝑓 ( 𝑥 ) = 0 . 0 0 3 𝑥 0 . 0 4 𝑥 .

  • A 𝑓 ( 𝑥 ) = 0 . 0 0 3 𝑥 0 . 0 4 𝑥 , 𝑓 ( 𝑥 ) = 0 . 0 0 3 𝑥 0 . 0 4 𝑥
  • B 𝑓 ( 𝑥 ) = 0 . 0 0 9 𝑥 0 . 1 6 𝑥 , 𝑓 ( 𝑥 ) = 0 . 0 2 7 𝑥 0 . 6 4 𝑥
  • C 𝑓 ( 𝑥 ) = 0 . 0 0 9 𝑥 0 . 1 6 𝑥 , 𝑓 ( 𝑥 ) = 0 . 0 3 6 𝑥 0 . 8 𝑥
  • D 𝑓 ( 𝑥 ) = 0 . 0 0 9 𝑥 0 . 1 6 𝑥 , 𝑓 ( 𝑥 ) = 0 . 0 1 8 𝑥 0 . 4 8 𝑥
  • E 𝑓 ( 𝑥 ) = 0 . 0 0 3 𝑥 0 . 0 4 𝑥 , 𝑓 ( 𝑥 ) = 0 . 0 0 3 𝑥 0 . 0 4 𝑥

Q12:

Given that 𝑦 = 4 𝑥 2 𝑥 + 4 2 𝑥 c o s s i n , find d d 𝑦 𝑥 at 𝑥 = 5 𝜋 2 .

  • A8
  • B0
  • C 4
  • D 4 0 𝜋

Q13:

Given that 𝑦 = 4 9 𝑥 5 t a n , determine 𝑦 .

  • A 3 6 5 9 𝑥 5 s e c
  • B 6 4 8 2 5 9 𝑥 5 9 𝑥 5 s e c t a n
  • C 6 4 8 2 5 9 𝑥 5 9 𝑥 5 s e c t a n
  • D 6 4 8 2 5 9 𝑥 5 9 𝑥 5 s e c t a n

Q14:

Find the third derivative of the function 𝑦 = 𝑥 + 5 𝑥 + 3 𝑥 + 2 𝑥 𝑥 9 .

  • A 2 0 𝑥 + 6 0 𝑥 + 1 8 𝑥
  • B 𝑥 + 5 𝑥 + 3
  • C 6 0 𝑥 + 1 2 0 𝑥 + 1 8 𝑥
  • D 6 0 𝑥 + 1 2 0 𝑥 + 1 8

Q15:

Given that 𝑦 = ( 4 𝑥 + 7 ) 7 𝑥 4 , determine d d 𝑦 𝑥 .

  • A 1 4 𝑥 4 9 𝑥
  • B 8 4 𝑥 9 8 𝑥 + 1 6
  • C 7 𝑥 4 9 𝑥 + 4
  • D 1 6 8 𝑥 9 8

Q16:

Find the second derivative of the function 𝑦 = 5 𝑥 4 2 𝑥 3 at the point ( 2 , 6 ) .

Q17:

Given 𝑦 = 𝑥 8 𝑥 8 and d d 𝑦 𝑥 9 𝑘 + 4 = 8 . Find the value of 𝑘 .

  • A 2
  • B 1 4 9
  • C6
  • D 2 3

Q18:

Find d d s i n 𝑥 ( 𝑥 ) by finding the first few derivatives and observing the pattern that occurs.

  • A 𝑥 s i n
  • B c o s 𝑥
  • C s i n 𝑥
  • D 𝑥 c o s
  • E 5 1 𝑥 𝑥 s i n c o s

Q19:

If 𝑦 = 𝑥 , find d d 𝑦 𝑥 .

  • A 𝑥 ( 8 ) !
  • B 9 !
  • C 8 !
  • D 𝑥 ( 9 ) !

Q20:

Evaluate d d d d s e c 𝑥 3 𝑥 + 𝑥 2 𝑥 9 𝑥 .

  • A 4 0 𝑥 9 𝑥 9 𝑥 𝑥 9 𝑥 t a n s e c s e c
  • B 4 0 𝑥 9 𝑥 9 𝑥 𝑥 + 9 𝑥 t a n s e c s e c
  • C 4 0 𝑥 9 𝑥 9 𝑥 𝑥 t a n s e c
  • D 4 0 𝑥 9 𝑥 9 𝑥 𝑥 9 𝑥 t a n s e c s e c

Q21:

Given that 𝑦 = 2 𝑥 5 , determine 𝑦 .

  • A 1 ( 2 𝑥 5 )
  • B 1 2 𝑥 5
  • C 3 8 ( 2 𝑥 5 )
  • D 3 ( 2 𝑥 5 )
  • E 3 ( 2 𝑥 5 )

Q22:

If 𝑦 𝑦 = 𝑥 1 𝑥 + 1 , find 𝑦 .

  • A 6 0 𝑥 4 0 𝑥 ( 𝑥 + 1 )
  • B 6 0 𝑥 4 0 𝑥 ( 𝑥 + 1 )
  • C 1 0 𝑥 ( 𝑥 + 1 )
  • D 6 0 𝑥 4 0 𝑥 ( 𝑥 + 1 )

Q23:

Given that 𝑦 = 6 𝑥 + 3 𝑥 7 𝑥 + 6 , determine d d 𝑦 𝑥 .

  • A 3 0 𝑥 + 6 𝑥 7 𝑥
  • B 3 0 𝑥 + 6 𝑥 7
  • C 6 𝑥 + 3 𝑥 7
  • D 6 2 0 𝑥 + 1

Q24:

Determine the second derivative of the function 𝑦 = 7 𝑥 + 3 𝑥 s i n c o s at 𝑥 = 𝜋 4 .

  • A 5 2
  • B 2 2
  • C 5 2
  • D 2 2

Q25:

Find the third derivative of the function 𝑦 = 3 𝑥 + 9 3 𝑥 s i n .

  • A 8 1 3 𝑥 + 6 s i n
  • B 2 4 3 3 𝑥 c o s
  • C 8 1 3 𝑥 + 6 s i n
  • D 2 4 3 3 𝑥 c o s
  • E 9 3 𝑥 c o s

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