Worksheet: Differentiation of Reciprocal Trigonometric Functions

In this worksheet, we will practice finding the derivatives of trigonometric functions, focusing on derivatives of cotangent, secant, and cosecant functions.

Q1:

If ๐‘ฆ = 6 ๐‘ฅ โˆ’ 7 ๐‘ฅ t a n c s c 2 , find d d ๐‘ฆ ๐‘ฅ at ๐‘ฅ = 3 ๐œ‹ 4 .

  • A โˆ’ 2
  • B40
  • C8
  • D โˆ’ 1 6

Q2:

Find d d ๐‘ฆ ๐‘ฅ , given that ๐‘ฅ = ๐‘ฆ ( 2 ๐‘ฅ โˆ’ 5 ) ๏Šจ s e c .

  • A 2 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 5 ) โˆ’ ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 5 ) c o s s i n ๏Šจ
  • B 2 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 5 ) + 2 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 5 ) c o s s i n ๏Šจ
  • C 2 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 5 ) + ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 5 ) c o s s i n ๏Šจ
  • D 2 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 5 ) โˆ’ 2 ๐‘ฅ ( 2 ๐‘ฅ โˆ’ 5 ) c o s s i n ๏Šจ

Q3:

Find d d ๐‘ฆ ๐‘ฅ if ๐‘ฆ = ๏€ผ 5 ๐œ‹ โˆ’ 3 7 ๐‘ฅ ๏ˆ c o t .

  • A 3 7 ๐‘ฅ ๏€ผ 5 ๐œ‹ โˆ’ 3 7 ๐‘ฅ ๏ˆ c s c ๏Šจ
  • B 3 7 ๐‘ฅ ๏€ผ 5 ๐œ‹ โˆ’ 3 7 ๐‘ฅ ๏ˆ ๏Šจ ๏Šจ c s c
  • C 3 7 ๐‘ฅ ๏€ผ 5 ๐œ‹ โˆ’ 3 7 ๐‘ฅ ๏ˆ ๏Šจ c s c
  • D โˆ’ 3 7 ๐‘ฅ ๏€ผ 5 ๐œ‹ โˆ’ 3 7 ๐‘ฅ ๏ˆ ๏Šจ ๏Šจ c s c
  • E โˆ’ 3 7 ๐‘ฅ ๏€ผ 5 ๐œ‹ โˆ’ 3 7 ๐‘ฅ ๏ˆ ๏Šจ c s c

Q4:

If ๐‘ฆ = ๐‘ฅ + 9 ๐‘ฅ s i n s e c , and ๐‘ฅ = 6 ๐œ‹ ๐‘ง , find d d ๐‘ฆ ๐‘ง at ๐‘ง = 4 .

  • A โˆ’ 6 ๐œ‹
  • B1
  • C 2 4 ๐œ‹
  • D 6 ๐œ‹

Q5:

Find for the function .

  • A
  • B
  • C
  • D

Q6:

If ๐‘ฆ = 8 ๐‘ฅ + 5 ๐‘ฅ c o t s e c , find d d ๐‘ฆ ๐‘ฅ at ๐‘ฅ = ๐œ‹ 6 .

  • A โˆ’ 3 2 + 5 โˆš 3 3
  • B โˆ’ 1 0 6 3
  • C โˆ’ 3 2 โˆ’ 1 0 โˆš 3
  • D โˆ’ 8 6 3

Q7:

Find d d ๐‘ฆ ๐‘ฅ if ๐‘ฆ = โˆ’ 5 7 ๐‘ฅ + 9 ๐‘ฅ s i n s e c ๏Šจ ๏Šจ .

  • A โˆ’ 5 7 ๐‘ฅ 7 ๐‘ฅ + 9 ๐‘ฅ ๐‘ฅ s i n c o s t a n s e c
  • B 2 ( โˆ’ 3 5 7 ๐‘ฅ + 9 ๐‘ฅ ๐‘ฅ ) s i n t a n s e c
  • C 7 7 ๐‘ฅ 7 ๐‘ฅ + ๐‘ฅ ๐‘ฅ s i n c o s t a n s e c
  • D 2 ( โˆ’ 3 5 7 ๐‘ฅ 7 ๐‘ฅ + 9 ๐‘ฅ ๐‘ฅ ) s i n c o s t a n s e c ๏Šจ
  • E โˆ’ 5 7 ๐‘ฅ + 9 ๐‘ฅ c o s t a n

Q8:

Given that ๐‘ฆ = ( 7 5 ๐‘ฅ + 3 6 ๐‘ฅ ) c o t c s c ๏Šฑ ๏Šง , find d d ๐‘ฆ ๐‘ฅ .

  • A โˆ’ 1 8 6 ๐‘ฅ 6 ๐‘ฅ + 3 5 5 ๐‘ฅ ( 7 5 ๐‘ฅ + 3 6 ๐‘ฅ ) c o t c s c c s c c o t c s c ๏Šจ ๏Šจ
  • B 1 8 6 ๐‘ฅ 6 ๐‘ฅ โˆ’ 3 5 5 ๐‘ฅ ( 7 5 ๐‘ฅ + 3 6 ๐‘ฅ ) c o t c s c c s c c o t c s c ๏Šจ ๏Šจ
  • C 3 6 ๐‘ฅ 6 ๐‘ฅ + 7 5 ๐‘ฅ ( 7 5 ๐‘ฅ + 3 6 ๐‘ฅ ) c o t c s c c s c c o t c s c ๏Šจ ๏Šจ
  • D 1 8 6 ๐‘ฅ 6 ๐‘ฅ + 3 5 5 ๐‘ฅ ( 7 5 ๐‘ฅ + 3 6 ๐‘ฅ ) c o t c s c c s c c o t c s c ๏Šจ ๏Šจ

Q9:

Find the derivative of the function ๐‘ฆ = ( ๐œƒ ) c o t s i n 2 .

  • A ๐‘ฆ โ€ฒ = โˆ’ 2 ๐œƒ ( ๐œƒ ) ( ๐œƒ ) c o s c o t s i n c s c s i n
  • B ๐‘ฆ โ€ฒ = 2 ๐œƒ ( ๐œƒ ) ( ๐œƒ ) c o s c o t s i n c s c s i n 2
  • C ๐‘ฆ โ€ฒ = ๐œƒ ( ๐œƒ ) ( ๐œƒ ) c o s c o t s i n c s c s i n 2
  • D ๐‘ฆ โ€ฒ = โˆ’ 2 ๐œƒ ( ๐œƒ ) ( ๐œƒ ) c o s c o t s i n c s c s i n 2
  • E ๐‘ฆ โ€ฒ = โˆ’ ๐œƒ ( ๐œƒ ) ( ๐œƒ ) c o s c o t s i n c s c s i n 2

Q10:

If ๐‘ฆ = ( ๐‘ฅ + 8 ๐‘ฅ ) ( ๐‘ฅ โˆ’ 8 ๐‘ฅ ) c s c c o t c s c c o t , find ๐‘ฆ โ€ฒ .

  • A โˆ’ ๐‘ฅ ๐‘ฅ + 8 8 ๐‘ฅ 8 ๐‘ฅ c o s c s c c o s c s c ๏Šจ ๏Šฉ ๏Šจ ๏Šฉ
  • B โˆ’ 2 ๐‘ฅ ๐‘ฅ + 1 6 8 ๐‘ฅ 8 ๐‘ฅ c o s c s c c o s c s c ๏Šจ ๏Šฉ ๏Šจ ๏Šฉ
  • C โˆ’ 2 ๐‘ฅ ๐‘ฅ + 2 8 ๐‘ฅ 8 ๐‘ฅ c o s c s c c o s c s c ๏Šฉ ๏Šฉ
  • D โˆ’ 2 ๐‘ฅ ๐‘ฅ + 1 6 8 ๐‘ฅ 8 ๐‘ฅ c o s c s c c o s c s c ๏Šฉ ๏Šฉ

Q11:

Given that ๐‘ฆ = 7 ๐‘ฅ ๏Žค ๏Žก + 2 c o t ๏€ฟ 1 โˆš ๐‘ฅ ๏‹ , find d ๐‘ฆ d ๐‘ฅ .

  • A 3 5 ๐‘ฅ โˆš ๐‘ฅ 2 โˆ’ 2 c s c ๏Šจ ๏€ฟ 1 โˆš ๐‘ฅ ๏‹
  • B 2 1 ๐‘ฅ โˆš ๐‘ฅ 2 โˆ’ 1 ๐‘ฅ โˆš ๐‘ฅ c s c ๏Šจ ๏€ฟ 1 โˆš ๐‘ฅ ๏‹
  • C 3 5 ๐‘ฅ โˆš ๐‘ฅ 2 + 1 โˆš ๐‘ฅ c s c ๏Šจ ๏€ฟ 1 โˆš ๐‘ฅ ๏‹
  • D 3 5 ๐‘ฅ โˆš ๐‘ฅ 2 + 1 ๐‘ฅ โˆš ๐‘ฅ c s c ๏Šจ ๏€ฟ 1 โˆš ๐‘ฅ ๏‹

Q12:

Given that ๐‘ฆ = 7 c o t 3 ๐‘ฅ 3 ๐‘ฅ โˆ’ 4 , find d ๐‘ฆ d ๐‘ฅ .

  • A โˆ’ ( 2 1 ๐‘ฅ โˆ’ 2 8 ) c s c ๏Šจ 3 ๐‘ฅ โˆ’ 2 1 c o t 3 ๐‘ฅ ( 3 ๐‘ฅ โˆ’ 4 ) ๏Šจ
  • B ( 6 3 ๐‘ฅ โˆ’ 8 4 ) c s c ๏Šจ 3 ๐‘ฅ โˆ’ 2 1 c o t 3 ๐‘ฅ ( 3 ๐‘ฅ โˆ’ 4 ) ๏Šจ
  • C ( 2 1 ๐‘ฅ โˆ’ 2 8 ) c s c ๏Šจ 3 ๐‘ฅ โˆ’ 2 1 c o t 3 ๐‘ฅ ( 3 ๐‘ฅ โˆ’ 4 ) ๏Šจ
  • D โˆ’ ( 6 3 ๐‘ฅ โˆ’ 8 4 ) c s c ๏Šจ 3 ๐‘ฅ โˆ’ 2 1 c o t 3 ๐‘ฅ ( 3 ๐‘ฅ โˆ’ 4 ) ๏Šจ
  • E โˆ’ ( 6 3 ๐‘ฅ โˆ’ 8 4 ) c s c ๏Šจ 3 ๐‘ฅ โˆ’ 7 c o t 3 ๐‘ฅ ( 3 ๐‘ฅ โˆ’ 4 ) ๏Šจ

Q13:

If ๐‘ฆ = 8 5 ๐‘ฅ โˆ’ 6 s e c ๏Šจ , find d d ๐‘ฆ ๐‘ฅ .

  • A 4 0 5 ๐‘ฅ 5 ๐‘ฅ t a n s e c ๏Šจ
  • B 1 6 5 ๐‘ฅ 5 ๐‘ฅ t a n s e c ๏Šจ
  • C 8 0 5 ๐‘ฅ s e c
  • D 8 0 5 ๐‘ฅ 5 ๐‘ฅ t a n s e c ๏Šจ
  • E 1 6 5 ๐‘ฅ s e c

Q14:

Given that ๐‘ฆ = 5 ๐‘ฅ 4 ๐‘ฅ ๏Šจ c o t , find d d ๐‘ฆ ๐‘ฅ .

  • A โˆ’ 2 0 ๐‘ฅ 4 ๐‘ฅ + 5 ๐‘ฅ 4 ๐‘ฅ ๏Šจ ๏Šจ c s c c o t
  • B 2 0 ๐‘ฅ 4 ๐‘ฅ + 1 0 ๐‘ฅ 4 ๐‘ฅ ๏Šจ ๏Šจ c s c c o t
  • C 2 0 ๐‘ฅ 4 ๐‘ฅ + 5 ๐‘ฅ 4 ๐‘ฅ ๏Šจ ๏Šจ c s c c o t
  • D โˆ’ 2 0 ๐‘ฅ 4 ๐‘ฅ + 1 0 ๐‘ฅ 4 ๐‘ฅ ๏Šจ ๏Šจ c s c c o t
  • E โˆ’ 5 ๐‘ฅ 4 ๐‘ฅ + 1 0 ๐‘ฅ 4 ๐‘ฅ ๏Šจ ๏Šจ c s c c o t

Q15:

If ๐‘ฆ = โˆš 1 9 ๐‘ฅ + 1 8 c s c , find d d ๐‘ฆ ๐‘ฅ .

  • A โˆ’ 1 9 ๐‘ฅ ๐‘ฅ 2 โˆš 1 9 ๐‘ฅ + 1 8 t a n c s c c s c
  • B 1 9 ๐‘ฅ ๐‘ฅ 2 โˆš 1 9 ๐‘ฅ + 1 8 c o t c s c c s c
  • C โˆ’ 1 9 ๐‘ฅ 2 โˆš 1 9 ๐‘ฅ + 1 8 t a n c s c
  • D โˆ’ 1 9 ๐‘ฅ ๐‘ฅ 2 โˆš 1 9 ๐‘ฅ + 1 8 c o t c s c c s c

Q16:

Find d d ๐‘ฆ ๐‘ฅ , given that ๐‘ฆ = โˆ’ 9 6 ๐‘ฅ โˆ’ 7 ๐‘ฅ t a n c s c .

  • A โˆ’ 7 7 ๐‘ฅ โˆ’ 5 4 6 ๐‘ฅ c o t s e c ๏Šจ ๏Šจ
  • B โˆ’ 7 7 ๐‘ฅ 7 ๐‘ฅ + 5 4 6 ๐‘ฅ c o t c s c s e c ๏Šจ
  • C 7 7 ๐‘ฅ โˆ’ 5 4 6 ๐‘ฅ c o t s e c
  • D 7 7 ๐‘ฅ 7 ๐‘ฅ โˆ’ 5 4 6 ๐‘ฅ c o t c s c s e c ๏Šจ
  • E โˆ’ 5 4 6 ๐‘ฅ + 7 7 ๐‘ฅ t a n c s c

Q17:

Given that ๐‘ฆ = โˆš 9 ๐‘ฅ + 5 3 ๐‘ฅ ๏Šจ ๏Šจ c o t , find d d ๐‘ฆ ๐‘ฅ .

  • A 1 8 ๐‘ฅ + 3 0 3 ๐‘ฅ 3 ๐‘ฅ 2 โˆš 9 ๐‘ฅ + 5 3 ๐‘ฅ c o t c s c c o t ๏Šจ ๏Šจ ๏Šจ
  • B 9 ๐‘ฅ โˆ’ 1 5 3 ๐‘ฅ 3 ๐‘ฅ 2 โˆš 9 ๐‘ฅ + 5 3 ๐‘ฅ c o t c s c c o t ๏Šจ ๏Šจ ๏Šจ
  • C 1 8 ๐‘ฅ + 1 0 3 ๐‘ฅ 3 ๐‘ฅ โˆš 9 ๐‘ฅ + 5 3 ๐‘ฅ c o t c s c c o t ๏Šจ ๏Šจ ๏Šจ
  • D 1 8 ๐‘ฅ โˆ’ 3 0 3 ๐‘ฅ 3 ๐‘ฅ 2 โˆš 9 ๐‘ฅ + 5 3 ๐‘ฅ c o t c s c c o t ๏Šจ ๏Šจ ๏Šจ

Q18:

Given that ๐‘ฆ = 3 ( ๐‘ฅ + 2 ) c s c ๏Šจ ๏Šซ , find d d ๐‘ฆ ๐‘ฅ .

  • A โˆ’ 6 ( ๐‘ฅ + 2 ) ( ๐‘ฅ + 2 ) c s c c o t ๏Šจ ๏Šซ ๏Šซ
  • B 3 0 ๐‘ฅ ( ๐‘ฅ + 2 ) ( ๐‘ฅ + 2 ) ๏Šช ๏Šจ ๏Šซ ๏Šซ c s c c o t
  • C 1 0 ( ๐‘ฅ + 2 ) ( ๐‘ฅ + 2 ) c s c c o t ๏Šจ ๏Šซ ๏Šซ
  • D โˆ’ 3 0 ๐‘ฅ ( ๐‘ฅ + 2 ) ( ๐‘ฅ + 2 ) ๏Šช ๏Šจ ๏Šซ ๏Šซ c s c c o t

Q19:

Given that ๐‘ฆ = 4 7 ( 8 ๐‘ฅ ) s e c t a n ๏Šจ , find d d ๐‘ฆ ๐‘ฅ .

  • A 8 7 ( 8 ๐‘ฅ ) ( 8 ๐‘ฅ ) t a n t a n s e c t a n ๏Šจ
  • B 4 7 ( 8 ๐‘ฅ ) ( 8 ๐‘ฅ ) ( 8 ๐‘ฅ ) t a n t a n s e c s e c t a n ๏Šจ ๏Šจ
  • C 4 7 ( 8 ๐‘ฅ ) ( 8 ๐‘ฅ ) t a n t a n s e c t a n ๏Šจ
  • D 6 4 7 ( 8 ๐‘ฅ ) ( 8 ๐‘ฅ ) ( 8 ๐‘ฅ ) t a n t a n s e c s e c t a n ๏Šจ ๏Šจ

Q20:

Differentiate ๐‘ฆ = ๐‘ฅ โˆ’ 3 ๐‘ฅ s e c c s c .

  • A ๐‘ฆ โ€ฒ = ๐‘ฅ ๐‘ฅ + 3 ๐‘ฅ ๐‘ฅ s e c c o t c s c t a n
  • B ๐‘ฆ โ€ฒ = โˆ’ ๐‘ฅ ๐‘ฅ โˆ’ 3 ๐‘ฅ ๐‘ฅ s e c t a n c s c c o t
  • C ๐‘ฆ โ€ฒ = ๐‘ฅ ๐‘ฅ โˆ’ 3 ๐‘ฅ ๐‘ฅ s e c t a n c s c c o t
  • D ๐‘ฆ โ€ฒ = ๐‘ฅ ๐‘ฅ + 3 ๐‘ฅ ๐‘ฅ s e c t a n c s c c o t
  • E ๐‘ฆ โ€ฒ = โˆ’ ๐‘ฅ ๐‘ฅ + 3 ๐‘ฅ ๐‘ฅ s e c t a n c s c c o t

Q21:

If ๐‘ฆ = โˆš โˆ’ 4 ๐‘ง + 9 , and ๐‘ง = 6 ๐‘ฅ s e c , find d d ๐‘ฆ ๐‘ฅ at ๐‘ฅ = ๐œ‹ 1 8 .

  • A 2 4 โˆš 3
  • B 6 โˆš 3
  • C 1 2
  • D โˆ’ 2 4 โˆš 3

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