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)=3𝑟5𝑟.

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

Q2:

Given that 𝑦=𝑎𝑥+𝑏𝑥, 𝑦=18, and 𝑦𝑥=14dd, find 𝑎 and 𝑏.

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

Q3:

Find the third derivative of the function 𝑦=44𝑥2𝑥sin.

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

Q4:

Given 𝑦=𝑥9, find dd𝑦𝑥.

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

Q5:

Given that 𝑦=(𝑥7)(4𝑥+7), and 𝑧=𝑥+5𝑥+9, determine dddd𝑦𝑥+𝑧𝑥.

Q6:

If 𝑓(𝑥)=𝑎𝑥+7𝑥8𝑥+9, and 𝑓(9)=9, find 𝑎.

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

Q7:

Determine the value of the second derivative of the function 𝑦=12𝑥8𝑥 at (1,4).

  • A 1 6
  • B16
  • C48
  • D 8

Q8:

If 𝑦=5𝑥sin, find 25𝑦𝑥+𝑦𝑥dddd.

Q9:

Given that 𝑦=3𝑥52𝑥+7, determine dd𝑦𝑥.

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

Q10:

Find the third derivative of the function 𝑦=11𝑥+14𝑥.

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

Q11:

Find the first and second derivatives of the function 𝑓(𝑥)=0.003𝑥0.04𝑥.

  • A 𝑓 ( 𝑥 ) = 0 . 0 0 9 𝑥 0 . 1 6 𝑥 , 𝑓 ( 𝑥 ) = 0 . 0 2 7 𝑥 0 . 6 4 𝑥
  • B 𝑓 ( 𝑥 ) = 0 . 0 0 9 𝑥 0 . 1 6 𝑥 , 𝑓 ( 𝑥 ) = 0 . 0 3 6 𝑥 0 . 8 𝑥
  • C 𝑓 ( 𝑥 ) = 0 . 0 0 3 𝑥 0 . 0 4 𝑥 , 𝑓 ( 𝑥 ) = 0 . 0 0 3 𝑥 0 . 0 4 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=4𝑥2𝑥+42𝑥cossin, find dd𝑦𝑥 at 𝑥=5𝜋2.

  • A0
  • B8
  • C 4 0 𝜋
  • D 4

Q13:

Given that 𝑦=49𝑥5tan, determine 𝑦.

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

Q14:

Find the third derivative of the function 𝑦=𝑥+5𝑥+3𝑥+2𝑥𝑥9.

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

Q15:

Given that 𝑦=(4𝑥+7)7𝑥4, determine dd𝑦𝑥.

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

Q16:

Find the second derivative of the function 𝑦=5𝑥42𝑥3 at the point (2,6).

Q17:

Given 𝑦=𝑥8𝑥8 and dd𝑦𝑥9𝑘+4=8. Find the value of 𝑘.

  • A 1 4 9
  • B 2 3
  • C 2
  • D6

Q18:

Find ddsin𝑥(𝑥) by finding the first few derivatives and observing the pattern that occurs.

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

Q19:

If 𝑦=𝑥, find dd𝑦𝑥.

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

Q20:

Evaluate ddddsec𝑥3𝑥+𝑥2𝑥9𝑥.

  • A 4 0 𝑥 9 𝑥 9 𝑥 𝑥 t a n 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 9 𝑥 t a n s e c 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=2𝑥5, determine 𝑦.

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

Q22:

If 𝑦𝑦=𝑥1𝑥+1, find 𝑦.

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

Q23:

Given that 𝑦=6𝑥+3𝑥7𝑥+6, determine dd𝑦𝑥.

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

Q24:

Determine the second derivative of the function 𝑦=7𝑥+3𝑥sincos at 𝑥=𝜋4.

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

Q25:

Find the third derivative of the function 𝑦=3𝑥+93𝑥sin.

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

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