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.

  • A 1 √ 1 − 𝑥 , where −1<𝑥<1
  • B − 1 √ 1 + 𝑥
  • C − 1 √ 1 − 𝑥 , where −1<𝑥<1
  • D 1 1 + 𝑥
  • E 1 √ 1 + 𝑥

Q2:

Find ddsin𝑥𝑥𝑎, where 𝑎≠0.

  • A 1 √ 𝑎 − 𝑥 , where |𝑥|<|𝑎|
  • B − 1 √ 𝑎 + 𝑥
  • C 1 √ 𝑎 + 𝑥
  • D − 1 √ 𝑎 − 𝑥 , where |𝑥|<|𝑎|
  • E 𝑎 𝑎 + 𝑥

Q3:

Find ddcsc𝑥𝑥𝑎.

  • A − 1 𝑥 √ 𝑥 − 𝑎 , where |𝑥|>|𝑎|
  • B 𝑎 𝑥 √ 𝑎 − 𝑥 , where |𝑥|<|𝑎| and 𝑥≠0
  • C − 𝑎 𝑥 √ 𝑎 − 𝑥 , where |𝑥|<|𝑎| and 𝑥≠0
  • D − 𝑎 | 𝑥 | √ 𝑥 − 𝑎 , where |𝑥|>|𝑎|
  • E 𝑎 | 𝑥 | √ 𝑥 − 𝑎 , where |𝑥|>|𝑎|

Q4:

Find ddcsc𝑥𝑥.

  • A 1 | 𝑥 | √ 𝑥 − 1
  • B − 1 𝑥 √ 𝑥 − 1
  • C − 1 𝑥 √ 1 − 𝑥
  • D 1 𝑥 √ 1 − 𝑥
  • E − 1 | 𝑥 | √ 𝑥 − 1

Q5:

Find ddsec𝑥𝑥𝑎, where 𝑎≠0.

  • A 𝑎 | 𝑥 | √ 𝑥 − 𝑎 , where |𝑥|>|𝑎|
  • B − 𝑎 𝑥 √ 𝑎 − 𝑥 , where 𝑥<|𝑎| and 𝑥≠0
  • C 𝑎 𝑥 √ 𝑥 − 𝑎 , where |𝑥|>|𝑎|
  • D − 𝑎 | 𝑥 | √ 𝑥 − 𝑎 , where |𝑥|>|𝑎|
  • E 𝑎 𝑥 √ 𝑎 − 𝑥 , where 𝑥<|𝑎| and 𝑥≠0

Q6:

Find ddsec𝑥𝑥.

  • A 1 𝑥 √ 𝑥 − 1 , where |𝑥|>1
  • B − 1 𝑥 √ 1 − 𝑥 , where −1<𝑥<1 and 𝑥≠0
  • C − 1 | 𝑥 | √ 𝑥 − 1 , where |𝑥|>1
  • D 1 | 𝑥 | √ 𝑥 − 1 , where |𝑥|>1
  • E 1 𝑥 √ 1 − 𝑥 , where −1<𝑥<1 and 𝑥≠0

Q7:

Find ddcot𝑥𝑥𝑎, where 𝑎≠0.

  • A − 𝑎 𝑎 + 𝑥
  • B 1 √ 𝑎 − 𝑥 , where |𝑥|<|𝑎|
  • C − 𝑎 𝑎 − 𝑥 , where |𝑥|<|𝑎|
  • D 𝑎 𝑎 − 𝑥 , where |𝑥|<|𝑎|
  • E 𝑎 𝑎 + 𝑥

Q8:

Find ddcot𝑥𝑥.

  • A − 1 1 + 𝑥
  • B − 1 1 − 𝑥 , where −1<𝑥<1
  • C 1 √ 1 − 𝑥 , where −1<𝑥<1
  • D 1 1 + 𝑥
  • E 1 1 − 𝑥 , where −1<𝑥<1

Q9:

Find ddtan𝑥𝑥𝑎, where 𝑎≠0.

  • A 𝑎 𝑎 + 𝑥
  • B − 𝑎 𝑎 + 𝑥
  • C 1 √ 𝑎 − 𝑥 , where |𝑥|<|𝑎|
  • D 𝑎 𝑎 − 𝑥 , where |𝑥|<|𝑎|
  • E − 𝑎 𝑎 − 𝑥 , where |𝑥|<|𝑎|

Q10:

Find ddtan𝑥𝑥.

  • A 1 1 + 𝑥
  • B − 1 1 + 𝑥
  • C 1 1 − 𝑥 , where −1<𝑥<1
  • D 1 √ 1 − 𝑥 , where −1<𝑥<1
  • E − 1 1 − 𝑥 , where −1<𝑥<1

Q11:

Find ddcos𝑥𝑥𝑎, where 𝑎≠0.

  • A 𝑎 𝑎 + 𝑥
  • B 1 √ 𝑎 − 𝑥 , where |𝑥|<|𝑎|
  • C 1 √ 𝑎 + 𝑥
  • D − 1 √ 𝑎 + 𝑥
  • E − 1 √ 𝑎 − 𝑥 , where |𝑥|<|𝑎|

Q12:

Find ddcos𝑥𝑥.

  • A 1 √ 1 + 𝑥
  • B 1 1 + 𝑥
  • C − 1 √ 1 − 𝑥 , where −1<𝑥<1
  • D 1 √ 1 − 𝑥 , where −1<𝑥<1
  • E − 1 √ 1 + 𝑥

Q13:

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

  • A d d 𝑦 𝑥 = 3 + 3 𝑥 𝑦 − 5 𝑦 − 5 𝑥 𝑦 − 8 𝑥 𝑦 4 𝑥 + 1 0 𝑥 𝑦 + 1 0 𝑥 𝑦
  • B d d 𝑦 𝑥 = 3 − 3 𝑥 𝑦 − 5 𝑦 − 5 𝑥 𝑦 − 8 𝑥 𝑦 4 𝑥 − 1 0 𝑥 𝑦 + 1 0 𝑥 𝑦
  • C d d 𝑦 𝑥 = 3 + 3 𝑥 𝑦 − 5 𝑦 − 5 𝑥 𝑦 + 4 𝑥 𝑦 1 0 𝑥 𝑦 + 1 0 𝑥 𝑦
  • D d d 𝑦 𝑥 = 3 + 3 𝑥 𝑦 − 5 𝑦 − 5 𝑥 𝑦 − 8 𝑥 𝑦 4 𝑥 + 5 𝑥 𝑦 + 5 𝑥 𝑦
  • E d d 𝑦 𝑥 = 3 + 3 𝑥 𝑦 − 5 𝑦 − 5 𝑥 𝑦 − 8 𝑥 𝑦 4 𝑥 − 1 0 𝑥 𝑦 − 1 0 𝑥 𝑦

Q14:

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

  • A 𝑎 1 + ( 𝑎 𝑥 )
  • B 𝑎 1 + 𝑎 𝑥
  • C 𝑎 𝑥 1 + ( 𝑎 𝑥 )
  • D 𝑎 √ 1 + ( 𝑎 𝑥 )
  • E 𝑎 𝑥 1 + 𝑎 𝑥

Q15:

Evaluate ddcot𝑥1𝑥.

  • A 1 𝑥 + 1
  • B 1 √ 𝑥 + 1 ( 𝑥 ) l n
  • C 1 √ 𝑥 + 1
  • D 1 𝑥 + 1 ( 𝑥 ) l n
  • E 𝑥 𝑥 + 1 ( 𝑥 ) l n

Q16:

Evaluate ddsin𝑥√1−𝑥.

  • A 𝑥 1 − 𝑥
  • B − 1 √ 1 − 𝑥
  • C1
  • D 1 √ 1 − 𝑥
  • E − 𝑥 1 − 𝑥

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