Worksheet: Derivatives of Inverse Trigonometric Functions

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

Q1:

Find ddsin𝑥𝑥.

  • A11𝑥, where 1<𝑥<1
  • B11+𝑥
  • C11𝑥, where 1<𝑥<1
  • D11+𝑥
  • E11+𝑥

Q2:

Find ddsin𝑥𝑥𝑎, where 𝑎0.

  • A1𝑎𝑥, where |𝑥|<|𝑎|
  • B1𝑎+𝑥
  • C1𝑎+𝑥
  • D1𝑎𝑥, where |𝑥|<|𝑎|
  • E𝑎𝑎+𝑥

Q3:

Find ddcsc𝑥𝑥𝑎.

  • A1𝑥𝑥𝑎, where |𝑥|>|𝑎|
  • B𝑎𝑥𝑎𝑥, where |𝑥|<|𝑎| and 𝑥0
  • C𝑎𝑥𝑎𝑥, where |𝑥|<|𝑎| and 𝑥0
  • D𝑎|𝑥|𝑥𝑎, where |𝑥|>|𝑎|
  • E𝑎|𝑥|𝑥𝑎, where |𝑥|>|𝑎|

Q4:

Find ddcsc𝑥𝑥.

  • A1|𝑥|𝑥1
  • B1𝑥𝑥1
  • C1𝑥1𝑥
  • D1𝑥1𝑥
  • E1|𝑥|𝑥1

Q5:

Find ddsec𝑥𝑥𝑎, where 𝑎0.

  • A𝑎|𝑥|𝑥𝑎, where |𝑥|>|𝑎|
  • B𝑎𝑥𝑎𝑥, where 𝑥<|𝑎| and 𝑥0
  • C𝑎𝑥𝑥𝑎, where |𝑥|>|𝑎|
  • D𝑎|𝑥|𝑥𝑎, where |𝑥|>|𝑎|
  • E𝑎𝑥𝑎𝑥, where 𝑥<|𝑎| and 𝑥0

Q6:

Find ddsec𝑥𝑥.

  • A1𝑥𝑥1, where |𝑥|>1
  • B1𝑥1𝑥, where 1<𝑥<1 and 𝑥0
  • C1|𝑥|𝑥1, where |𝑥|>1
  • D1|𝑥|𝑥1, where |𝑥|>1
  • E1𝑥1𝑥, where 1<𝑥<1 and 𝑥0

Q7:

Find ddcot𝑥𝑥𝑎, where 𝑎0.

  • A𝑎𝑎+𝑥
  • B1𝑎𝑥, where |𝑥|<|𝑎|
  • C𝑎𝑎𝑥, where |𝑥|<|𝑎|
  • D𝑎𝑎𝑥, where |𝑥|<|𝑎|
  • E𝑎𝑎+𝑥

Q8:

Find ddcot𝑥𝑥.

  • A11+𝑥
  • B11𝑥, where 1<𝑥<1
  • C11𝑥, where 1<𝑥<1
  • D11+𝑥
  • E11𝑥, where 1<𝑥<1

Q9:

Find ddtan𝑥𝑥𝑎, where 𝑎0.

  • A𝑎𝑎+𝑥
  • B𝑎𝑎+𝑥
  • C1𝑎𝑥, where |𝑥|<|𝑎|
  • D𝑎𝑎𝑥, where |𝑥|<|𝑎|
  • E𝑎𝑎𝑥, where |𝑥|<|𝑎|

Q10:

Find ddtan𝑥𝑥.

  • A11+𝑥
  • B11+𝑥
  • C11𝑥, where 1<𝑥<1
  • D11𝑥, where 1<𝑥<1
  • E11𝑥, where 1<𝑥<1

Q11:

Find ddcos𝑥𝑥𝑎, where 𝑎0.

  • A𝑎𝑎+𝑥
  • B1𝑎𝑥, where |𝑥|<|𝑎|
  • C1𝑎+𝑥
  • D1𝑎+𝑥
  • E1𝑎𝑥, where |𝑥|<|𝑎|

Q12:

Find ddcos𝑥𝑥.

  • A11+𝑥
  • B11+𝑥
  • C11𝑥, where 1<𝑥<1
  • D11𝑥, where 1<𝑥<1
  • E11+𝑥

Q13:

Given that 4𝑥𝑦=3𝑥5𝑥𝑦tan, find dd𝑦𝑥 by implicit differentiation.

  • Add𝑦𝑥=3+3𝑥𝑦5𝑦5𝑥𝑦8𝑥𝑦4𝑥+10𝑥𝑦+10𝑥𝑦
  • Bdd𝑦𝑥=33𝑥𝑦5𝑦5𝑥𝑦8𝑥𝑦4𝑥10𝑥𝑦+10𝑥𝑦
  • Cdd𝑦𝑥=3+3𝑥𝑦5𝑦5𝑥𝑦+4𝑥𝑦10𝑥𝑦+10𝑥𝑦
  • Ddd𝑦𝑥=3+3𝑥𝑦5𝑦5𝑥𝑦8𝑥𝑦4𝑥+5𝑥𝑦+5𝑥𝑦
  • Edd𝑦𝑥=3+3𝑥𝑦5𝑦5𝑥𝑦8𝑥𝑦4𝑥10𝑥𝑦10𝑥𝑦

Q14:

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

  • A𝑎1+(𝑎𝑥)
  • B𝑎1+𝑎𝑥
  • C𝑎𝑥1+(𝑎𝑥)
  • D𝑎1+(𝑎𝑥)
  • E𝑎𝑥1+𝑎𝑥

Q15:

Evaluate ddcot𝑥1𝑥.

  • A1𝑥+1
  • B1𝑥+1(𝑥)ln
  • C1𝑥+1
  • D1𝑥+1(𝑥)ln
  • E𝑥𝑥+1(𝑥)ln

Q16:

Evaluate ddsin𝑥1𝑥.

  • A𝑥1𝑥
  • B11𝑥
  • C1
  • D11𝑥
  • E𝑥1𝑥

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