Worksheet: Derivatives of Inverse Trigonometric Functions

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

Q1:

Find ddsin𝑥𝑥.

  • A1√1−𝑥, where −1<𝑥<1
  • B−1√1+𝑥
  • C−1√1−𝑥, where −1<𝑥<1
  • D11+𝑥
  • E1√1+𝑥

Q2:

Find ddsin𝑥𝑥𝑎, where 𝑎≠0.

  • A1√𝑎−𝑥, where |𝑥|<|𝑎|
  • B−1√𝑎+𝑥
  • C1√𝑎+𝑥
  • D−1√𝑎−𝑥, where |𝑥|<|𝑎|
  • E𝑎𝑎+𝑥

Q3:

Find ddcsc𝑥𝑥𝑎.

  • A−1𝑥√𝑥−𝑎, where |𝑥|>|𝑎|
  • B𝑎𝑥√𝑎−𝑥, where |𝑥|<|𝑎| and 𝑥≠0
  • C−𝑎𝑥√𝑎−𝑥, where |𝑥|<|𝑎| and 𝑥≠0
  • D−𝑎|𝑥|√𝑥−𝑎, where |𝑥|>|𝑎|
  • E𝑎|𝑥|√𝑥−𝑎, where |𝑥|>|𝑎|

Q4:

Find ddcsc𝑥𝑥.

  • A1|𝑥|√𝑥−1
  • B−1𝑥√𝑥−1
  • C−1𝑥√1−𝑥
  • D1𝑥√1−𝑥
  • E−1|𝑥|√𝑥−1

Q5:

Find ddsec𝑥𝑥𝑎, where 𝑎≠0.

  • A𝑎|𝑥|√𝑥−𝑎, where |𝑥|>|𝑎|
  • B−𝑎𝑥√𝑎−𝑥, where 𝑥<|𝑎| and 𝑥≠0
  • C𝑎𝑥√𝑥−𝑎, where |𝑥|>|𝑎|
  • D−𝑎|𝑥|√𝑥−𝑎, where |𝑥|>|𝑎|
  • E𝑎𝑥√𝑎−𝑥, where 𝑥<|𝑎| and 𝑥≠0

Q6:

Find ddsec𝑥𝑥.

  • A1𝑥√𝑥−1, where |𝑥|>1
  • B−1𝑥√1−𝑥, where −1<𝑥<1 and 𝑥≠0
  • C−1|𝑥|√𝑥−1, where |𝑥|>1
  • D1|𝑥|√𝑥−1, where |𝑥|>1
  • E1𝑥√1−𝑥, where −1<𝑥<1 and 𝑥≠0

Q7:

Find ddcot𝑥𝑥𝑎, where 𝑎≠0.

  • A−𝑎𝑎+𝑥
  • B1√𝑎−𝑥, where |𝑥|<|𝑎|
  • C−𝑎𝑎−𝑥, where |𝑥|<|𝑎|
  • D𝑎𝑎−𝑥, where |𝑥|<|𝑎|
  • E𝑎𝑎+𝑥

Q8:

Find ddcot𝑥𝑥.

  • A−11+𝑥
  • B−11−𝑥, where −1<𝑥<1
  • C1√1−𝑥, where −1<𝑥<1
  • D11+𝑥
  • E11−𝑥, where −1<𝑥<1

Q9:

Find ddtan𝑥𝑥𝑎, where 𝑎≠0.

  • A𝑎𝑎+𝑥
  • B−𝑎𝑎+𝑥
  • C1√𝑎−𝑥, where |𝑥|<|𝑎|
  • D𝑎𝑎−𝑥, where |𝑥|<|𝑎|
  • E−𝑎𝑎−𝑥, where |𝑥|<|𝑎|

Q10:

Find ddtan𝑥𝑥.

  • A11+𝑥
  • B−11+𝑥
  • C11−𝑥, where −1<𝑥<1
  • D1√1−𝑥, where −1<𝑥<1
  • E−11−𝑥, where −1<𝑥<1

Q11:

Find ddcos𝑥𝑥𝑎, where 𝑎≠0.

  • A𝑎𝑎+𝑥
  • B1√𝑎−𝑥, where |𝑥|<|𝑎|
  • C1√𝑎+𝑥
  • D−1√𝑎+𝑥
  • E−1√𝑎−𝑥, where |𝑥|<|𝑎|

Q12:

Find ddcos𝑥𝑥.

  • A1√1+𝑥
  • B11+𝑥
  • C−1√1−𝑥, where −1<𝑥<1
  • D1√1−𝑥, where −1<𝑥<1
  • E−1√1+𝑥

Q13:

Given that 4𝑥𝑦=3𝑥−5𝑥𝑦tan, find dd𝑦𝑥 by implicit differentiation.

  • Add𝑦𝑥=3+3𝑥𝑦−5𝑦−5𝑥𝑦−8𝑥𝑦4𝑥+10𝑥𝑦+10𝑥𝑦
  • Bdd𝑦𝑥=3−3𝑥𝑦−5𝑦−5𝑥𝑦−8𝑥𝑦4𝑥−10𝑥𝑦+10𝑥𝑦
  • Cdd𝑦𝑥=3+3𝑥𝑦−5𝑦−5𝑥𝑦+4𝑥𝑦10𝑥𝑦+10𝑥𝑦
  • Ddd𝑦𝑥=3+3𝑥𝑦−5𝑦−5𝑥𝑦−8𝑥𝑦4𝑥+5𝑥𝑦+5𝑥𝑦
  • Edd𝑦𝑥=3+3𝑥𝑦−5𝑦−5𝑥𝑦−8𝑥𝑦4𝑥−10𝑥𝑦−10𝑥𝑦

Q14:

Find an expression for the derivative of 𝑦=𝑎𝑥tan in terms of 𝑥.

  • A𝑎1+(𝑎𝑥)
  • B𝑎1+𝑎𝑥
  • C𝑎𝑥1+(𝑎𝑥)
  • D𝑎√1+(𝑎𝑥)
  • E𝑎𝑥1+𝑎𝑥

Q15:

Evaluate ddcot𝑥1𝑥.

  • A1𝑥+1
  • B1√𝑥+1(𝑥)ln
  • C1√𝑥+1
  • D1𝑥+1(𝑥)ln
  • E𝑥𝑥+1(𝑥)ln

Q16:

Evaluate ddsin𝑥√1−𝑥.

  • A𝑥1−𝑥
  • B−1√1−𝑥
  • C1
  • D1√1−𝑥
  • E−𝑥1−𝑥

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