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 𝑦=6𝑥7𝑥tancsc, find dd𝑦𝑥 at 𝑥=3𝜋4.

  • A 1 6
  • B40
  • C 2
  • D8

Q2:

Find dd𝑦𝑥, given that 𝑥=𝑦(2𝑥5)sec.

  • A 2 𝑥 ( 2 𝑥 5 ) + 𝑥 ( 2 𝑥 5 ) c o s s i n
  • B 2 𝑥 ( 2 𝑥 5 ) 𝑥 ( 2 𝑥 5 ) c o s s i n
  • C 2 𝑥 ( 2 𝑥 5 ) 2 𝑥 ( 2 𝑥 5 ) c o s s i n
  • D 2 𝑥 ( 2 𝑥 5 ) + 2 𝑥 ( 2 𝑥 5 ) c o s s i n

Q3:

Find dd𝑦𝑥 if 𝑦=5𝜋37𝑥cot.

  • A 3 7 𝑥 5 𝜋 3 7 𝑥 c s c
  • B 3 7 𝑥 5 𝜋 3 7 𝑥 c s c
  • C 3 7 𝑥 5 𝜋 3 7 𝑥 c s c
  • D 3 7 𝑥 5 𝜋 3 7 𝑥 c s c
  • E 3 7 𝑥 5 𝜋 3 7 𝑥 c s c

Q4:

If 𝑦=𝑥+9𝑥sinsec, and 𝑥=6𝜋𝑧, find dd𝑦𝑧 at 𝑧=4.

  • A1
  • B 6 𝜋
  • C 2 4 𝜋
  • D 6 𝜋

Q5:

Find dd𝑦𝑥 for the function 𝑦=34𝑥3csc.

  • A 6 0 𝑥 4 𝑥 3 4 𝑥 3 c o t c s c
  • B 6 0 𝑥 2 0 𝑥 2 0 𝑥 c o t c s c
  • C 6 0 𝑥 4 𝑥 3 4 𝑥 3 c o t c s c
  • D 6 0 4 𝑥 3 4 𝑥 3 c o t c s c

Q6:

If 𝑦=8𝑥+5𝑥cotsec, find dd𝑦𝑥 at 𝑥=𝜋6 .

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

Q7:

Find dd𝑦𝑥 if 𝑦=57𝑥+9𝑥sinsec.

  • A 2 ( 3 5 7 𝑥 7 𝑥 + 9 𝑥 𝑥 ) s i n c o s t a n s e c
  • B 7 7 𝑥 7 𝑥 + 𝑥 𝑥 s i n c o s t a n s e c
  • C 5 7 𝑥 + 9 𝑥 c o s t a n
  • D 5 7 𝑥 7 𝑥 + 9 𝑥 𝑥 s i n c o s t a n s e c
  • E 2 ( 3 5 7 𝑥 + 9 𝑥 𝑥 ) s i n t a n s e c

Q8:

Given that 𝑦=(75𝑥+36𝑥)cotcsc, find dd𝑦𝑥.

  • A 3 6 𝑥 6 𝑥 + 7 5 𝑥 ( 7 5 𝑥 + 3 6 𝑥 ) c o t c s c c s c c o t c s c
  • B 1 8 6 𝑥 6 𝑥 + 3 5 5 𝑥 ( 7 5 𝑥 + 3 6 𝑥 ) c o t c s c c s c c o t c s c
  • C 1 8 6 𝑥 6 𝑥 + 3 5 5 𝑥 ( 7 5 𝑥 + 3 6 𝑥 ) c o t c s c c s c c o t c s c
  • D 1 8 6 𝑥 6 𝑥 3 5 5 𝑥 ( 7 5 𝑥 + 3 6 𝑥 ) c o t c s c c s c c o t c s c

Q9:

Find the derivative of the function 𝑦=(𝜃)cotsin.

  • A 𝑦 = 2 𝜃 ( 𝜃 ) ( 𝜃 ) c o s c o t s i n c s c s i n
  • B 𝑦 = 2 𝜃 ( 𝜃 ) ( 𝜃 ) c o s c o t s i n c s c s i n
  • C 𝑦 = 2 𝜃 ( 𝜃 ) ( 𝜃 ) c o s c o t s i n c s c s i n
  • D 𝑦 = 𝜃 ( 𝜃 ) ( 𝜃 ) c o s c o t s i n c s c s i n
  • E 𝑦 = 𝜃 ( 𝜃 ) ( 𝜃 ) c o s c o t s i n c s c s i n

Q10:

If 𝑦=(𝑥+8𝑥)(𝑥8𝑥)csccotcsccot, find 𝑦.

  • A 𝑥 𝑥 + 8 8 𝑥 8 𝑥 c o s c s c c o s c s c
  • B 2 𝑥 𝑥 + 1 6 8 𝑥 8 𝑥 c o s c s c c o s c s c
  • C 2 𝑥 𝑥 + 2 8 𝑥 8 𝑥 c o s c s c c o s c s c
  • D 2 𝑥 𝑥 + 1 6 8 𝑥 8 𝑥 c o s c s c c o s c s c

Q11:

Given that 𝑦=7𝑥+21𝑥cot, find dd𝑦𝑥.

  • A 2 1 𝑥 𝑥 2 1 𝑥 𝑥 1 𝑥 c s c
  • B 3 5 𝑥 𝑥 2 + 1 𝑥 1 𝑥 c s c
  • C 3 5 𝑥 𝑥 2 + 1 𝑥 𝑥 1 𝑥 c s c
  • D 3 5 𝑥 𝑥 2 2 1 𝑥 c s c

Q12:

Given that 𝑦=73𝑥3𝑥4cot, find dd𝑦𝑥.

  • A ( 6 3 𝑥 8 4 ) 3 𝑥 2 1 3 𝑥 ( 3 𝑥 4 ) c s c c o t
  • B ( 6 3 𝑥 8 4 ) 3 𝑥 7 3 𝑥 ( 3 𝑥 4 ) c s c c o t
  • C ( 2 1 𝑥 2 8 ) 3 𝑥 2 1 3 𝑥 ( 3 𝑥 4 ) c s c c o t
  • D ( 2 1 𝑥 2 8 ) 3 𝑥 2 1 3 𝑥 ( 3 𝑥 4 ) c s c c o t
  • E ( 6 3 𝑥 8 4 ) 3 𝑥 2 1 3 𝑥 ( 3 𝑥 4 ) c s c c o t

Q13:

If 𝑦=85𝑥6sec, find dd𝑦𝑥.

  • A 1 6 5 𝑥 5 𝑥 t a n s e c
  • B 1 6 5 𝑥 s e c
  • C 4 0 5 𝑥 5 𝑥 t a n s e c
  • D 8 0 5 𝑥 5 𝑥 t a n s e c
  • E 8 0 5 𝑥 s e c

Q14:

Given that 𝑦=5𝑥4𝑥cot, find dd𝑦𝑥.

  • A 2 0 𝑥 4 𝑥 + 1 0 𝑥 4 𝑥 c s c c o t
  • B 5 𝑥 4 𝑥 + 1 0 𝑥 4 𝑥 c s c c o t
  • C 2 0 𝑥 4 𝑥 + 1 0 𝑥 4 𝑥 c s c c o t
  • D 2 0 𝑥 4 𝑥 + 5 𝑥 4 𝑥 c s c c o t
  • E 2 0 𝑥 4 𝑥 + 5 𝑥 4 𝑥 c s c c o t

Q15:

If 𝑦=19𝑥+18csc, find dd𝑦𝑥.

  • A 1 9 𝑥 𝑥 2 1 9 𝑥 + 1 8 c o t c s c c s c
  • B 1 9 𝑥 𝑥 2 1 9 𝑥 + 1 8 c o t c s c c s c
  • C 1 9 𝑥 2 1 9 𝑥 + 1 8 t a n c s c
  • D 1 9 𝑥 𝑥 2 1 9 𝑥 + 1 8 t a n c s c c s c

Q16:

Find dd𝑦𝑥, given that 𝑦=96𝑥7𝑥tancsc.

  • A 5 4 6 𝑥 + 7 7 𝑥 t a n c s c
  • B 7 7 𝑥 7 𝑥 + 5 4 6 𝑥 c o t c s c s e c
  • C 7 7 𝑥 5 4 6 𝑥 c o t s e c
  • D 7 7 𝑥 5 4 6 𝑥 c o t s e c
  • E 7 7 𝑥 7 𝑥 5 4 6 𝑥 c o t c s c s e c

Q17:

Given that 𝑦=9𝑥+53𝑥cot, find dd𝑦𝑥.

  • A 9 𝑥 1 5 3 𝑥 3 𝑥 2 9 𝑥 + 5 3 𝑥 c o t c s c c o t
  • B 1 8 𝑥 + 3 0 3 𝑥 3 𝑥 2 9 𝑥 + 5 3 𝑥 c o t c s c c o t
  • C 1 8 𝑥 3 0 3 𝑥 3 𝑥 2 9 𝑥 + 5 3 𝑥 c o t c s c c o t
  • D 1 8 𝑥 + 1 0 3 𝑥 3 𝑥 9 𝑥 + 5 3 𝑥 c o t c s c c o t

Q18:

Given that 𝑦=3(𝑥+2)csc, find dd𝑦𝑥.

  • A 3 0 𝑥 ( 𝑥 + 2 ) ( 𝑥 + 2 ) c s c c o t
  • B 3 0 𝑥 ( 𝑥 + 2 ) ( 𝑥 + 2 ) c s c c o t
  • C 1 0 ( 𝑥 + 2 ) ( 𝑥 + 2 ) c s c c o t
  • D 6 ( 𝑥 + 2 ) ( 𝑥 + 2 ) c s c c o t

Q19:

Given that 𝑦=47(8𝑥)sectan, find dd𝑦𝑥.

  • A 8 7 ( 8 𝑥 ) ( 8 𝑥 ) t a n t a n s e c t a n
  • B 4 7 ( 8 𝑥 ) ( 8 𝑥 ) ( 8 𝑥 ) t a n t a n s e c s e c t a n
  • C 6 4 7 ( 8 𝑥 ) ( 8 𝑥 ) ( 8 𝑥 ) t a n t a n s e c s e c t a n
  • D 4 7 ( 8 𝑥 ) ( 8 𝑥 ) t a n t a n s e c t a n

Q20:

Differentiate 𝑦=𝑥3𝑥seccsc.

  • A 𝑦 = 𝑥 𝑥 3 𝑥 𝑥 s e c t a n c s c c o t
  • B 𝑦 = 𝑥 𝑥 + 3 𝑥 𝑥 s e c c o t c s c t a n
  • C 𝑦 = 𝑥 𝑥 + 3 𝑥 𝑥 s e c t a n c s c c o t
  • D 𝑦 = 𝑥 𝑥 + 3 𝑥 𝑥 s e c t a n c s c c o t
  • E 𝑦 = 𝑥 𝑥 3 𝑥 𝑥 s e c t a n c s c c o t

Q21:

If 𝑦=4𝑧+9, and 𝑧=6𝑥sec, find dd𝑦𝑥 at 𝑥=𝜋18.

  • A 2 4 3
  • B 1 2
  • C 6 3
  • D 2 4 3

Q22:

Find the equation of the tangent to the curve 𝑦=7𝑥3𝑥cossec at 𝑥=𝜋6.

  • A 𝑦 + 1 1 𝑥 2 3 3 2 + 𝜋 6 = 0
  • B 𝑦 + 1 1 𝑥 2 1 1 𝜋 1 2 + 3 3 2 = 0
  • C 𝑦 1 1 𝑥 2 3 3 2 + 1 1 𝜋 1 2 = 0
  • D 𝑦 + 1 1 𝑥 2 1 1 𝜋 1 2 3 3 2 = 0

Q23:

If 𝑦=98𝑥8𝑥tansec, find dd𝑦𝑥.

  • A 7 2 8 𝑥 8 𝑥 7 2 8 𝑥 t a n s e c s e c
  • B 9 8 𝑥 8 𝑥 9 8 𝑥 t a n s e c s e c
  • C 9 8 𝑥 9 8 𝑥 t a n s e c
  • D 7 2 8 𝑥 7 2 8 𝑥 t a n s e c

Q24:

Find dd𝑦𝑥, given that 𝑦=(35𝑥+7)cot.

  • A 6 0 ( 3 5 𝑥 + 7 ) 5 𝑥 c o t c s c
  • B 6 0 ( 3 5 𝑥 + 7 ) c o t
  • C 4 ( 3 5 𝑥 + 7 ) 5 𝑥 c o t c s c
  • D 6 0 ( 3 5 𝑥 + 7 ) 5 𝑥 c o t c s c
  • E 2 0 ( 3 5 𝑥 + 7 ) c o t

Q25:

Given 𝑦=(𝑥+3)(9𝑥+𝑥)csc, find dd𝑦𝑥.

  • A 1 8 𝑥 + ( 𝑥 + 3 ) 𝑥 𝑥 + 𝑥 + 2 7 c o t c s c c s c
  • B 9 𝑥 ( 𝑥 + 3 ) 𝑥 𝑥 + 𝑥 + 2 7 c o t c s c c s c
  • C 1 8 𝑥 ( 𝑥 + 3 ) 𝑥 𝑥 + 𝑥 + 2 7 c o t c s c c s c
  • D 1 8 𝑥 ( 𝑥 3 ) 𝑥 𝑥 + 𝑥 + 2 7 c o t c s c c s c

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.