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.

  • A𝐺(𝑟)=32𝑟𝑟, 𝐺(𝑟)=34𝑟+45𝑟
  • B𝐺(𝑟)=3𝑟5𝑟, 𝐺(𝑟)=3𝑟5𝑟
  • C𝐺(𝑟)=32𝑟𝑟, 𝐺(𝑟)=34𝑟+45𝑟
  • D𝐺(𝑟)=3𝑟5𝑟, 𝐺(𝑟)=32𝑟+4𝑟
  • E𝐺(𝑟)=32𝑟𝑟, 𝐺(𝑟)=34𝑟+45𝑟

Q2:

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

  • A𝑎=6,𝑏=43
  • B𝑎=3,𝑏=25
  • C𝑎=3,𝑏=11
  • D𝑎=6,𝑏=29

Q3:

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

  • A176𝑥2𝑥+1762𝑥sincos
  • B8𝑥2𝑥cos
  • C352𝑥2𝑥5282𝑥cossin
  • D352𝑥2𝑥+5282𝑥cossin
  • E176𝑥2𝑥1762𝑥sincos

Q4:

Given 𝑦=𝑥9, find dd𝑦𝑥.

  • A34(𝑥9)
  • B14(𝑥9)
  • C4(𝑥9)
  • D12(𝑥9)

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.

  • A16
  • B169
  • C2354
  • D89

Q7:

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

Q8:

If 𝑦=5𝑥sin, find 25𝑦𝑥+𝑦𝑥dddd.

Q9:

Given that 𝑦=3𝑥52𝑥+7, determine dd𝑦𝑥.

  • A6276𝑥(2𝑥+7)
  • B62𝑥(2𝑥+7)
  • C6276𝑥(2𝑥+7)
  • D76𝑥(2𝑥+7)
  • E627+6𝑥(2𝑥+7)

Q10:

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

  • A84𝑥
  • B14𝑥
  • C28𝑥
  • D84𝑥

Q11:

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

  • A𝑓(𝑥)=0.009𝑥0.16𝑥, 𝑓(𝑥)=0.027𝑥0.64𝑥
  • B𝑓(𝑥)=0.009𝑥0.16𝑥, 𝑓(𝑥)=0.036𝑥0.8𝑥
  • C𝑓(𝑥)=0.003𝑥0.04𝑥, 𝑓(𝑥)=0.003𝑥0.04𝑥
  • D𝑓(𝑥)=0.009𝑥0.16𝑥, 𝑓(𝑥)=0.018𝑥0.48𝑥
  • E𝑓(𝑥)=0.003𝑥0.04𝑥, 𝑓(𝑥)=0.003𝑥0.04𝑥

Q12:

Given that 𝑦=4𝑥2𝑥+42𝑥cossin, find dd𝑦𝑥 at 𝑥=5𝜋2.

  • A0
  • B8
  • C40𝜋
  • D4

Q13:

Given that 𝑦=49𝑥5tan, determine 𝑦.

  • A648259𝑥59𝑥5sectan
  • B648259𝑥59𝑥5sectan
  • C648259𝑥59𝑥5sectan
  • D3659𝑥5sec

Q14:

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

  • A60𝑥+120𝑥+18
  • B20𝑥+60𝑥+18𝑥
  • C60𝑥+120𝑥+18𝑥
  • D𝑥+5𝑥+3

Q15:

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

  • A14𝑥49𝑥
  • B7𝑥49𝑥+4
  • C84𝑥98𝑥+16
  • D168𝑥98

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.

  • A149
  • B23
  • C2
  • D6

Q18:

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

  • A51𝑥𝑥sincos
  • B𝑥cos
  • Csin𝑥
  • Dcos𝑥
  • E𝑥sin

Q19:

If 𝑦=𝑥, find dd𝑦𝑥.

  • A𝑥(8)!
  • B𝑥(9)!
  • C8!
  • D9!

Q20:

Evaluate ddddsec𝑥3𝑥+𝑥2𝑥9𝑥.

  • A40𝑥9𝑥9𝑥𝑥tansec
  • B40𝑥9𝑥9𝑥𝑥+9𝑥tansecsec
  • C40𝑥9𝑥9𝑥𝑥9𝑥tansecsec
  • D40𝑥9𝑥9𝑥𝑥9𝑥tansecsec

Q21:

Given that 𝑦=2𝑥5, determine 𝑦.

  • A1(2𝑥5)
  • B3(2𝑥5)
  • C12𝑥5
  • D3(2𝑥5)
  • E38(2𝑥5)

Q22:

If 𝑦𝑦=𝑥1𝑥+1, find 𝑦.

  • A60𝑥40𝑥(𝑥+1)
  • B10𝑥(𝑥+1)
  • C60𝑥40𝑥(𝑥+1)
  • D60𝑥40𝑥(𝑥+1)

Q23:

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

  • A620𝑥+1
  • B30𝑥+6𝑥7𝑥
  • C30𝑥+6𝑥7
  • D6𝑥+3𝑥7

Q24:

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

  • A52
  • B22
  • C52
  • D22

Q25:

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

  • A813𝑥+6sin
  • B93𝑥cos
  • C2433𝑥cos
  • D813𝑥+6sin
  • E2433𝑥cos

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