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.

  • A16
  • B40
  • C2
  • D8

Q2:

Find dd𝑦𝑥, given that 𝑥=𝑦(2𝑥5)sec.

  • A2𝑥(2𝑥5)+𝑥(2𝑥5)cossin
  • B2𝑥(2𝑥5)𝑥(2𝑥5)cossin
  • C2𝑥(2𝑥5)2𝑥(2𝑥5)cossin
  • D2𝑥(2𝑥5)+2𝑥(2𝑥5)cossin

Q3:

Find dd𝑦𝑥 if 𝑦=5𝜋37𝑥cot.

  • A37𝑥5𝜋37𝑥csc
  • B37𝑥5𝜋37𝑥csc
  • C37𝑥5𝜋37𝑥csc
  • D37𝑥5𝜋37𝑥csc
  • E37𝑥5𝜋37𝑥csc

Q4:

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

  • A1
  • B6𝜋
  • C24𝜋
  • D6𝜋

Q5:

Find dd𝑦𝑥 for the function 𝑦=34𝑥3csc.

  • A60𝑥4𝑥34𝑥3cotcsc
  • B60𝑥20𝑥20𝑥cotcsc
  • C60𝑥4𝑥34𝑥3cotcsc
  • D604𝑥34𝑥3cotcsc

Q6:

If 𝑦=8𝑥+5𝑥cotsec, find dd𝑦𝑥 at 𝑥=𝜋6 .

  • A863
  • B32+533
  • C1063
  • D32103

Q7:

Find dd𝑦𝑥 if 𝑦=57𝑥+9𝑥sinsec.

  • A2(357𝑥7𝑥+9𝑥𝑥)sincostansec
  • B77𝑥7𝑥+𝑥𝑥sincostansec
  • C57𝑥+9𝑥costan
  • D57𝑥7𝑥+9𝑥𝑥sincostansec
  • E2(357𝑥+9𝑥𝑥)sintansec

Q8:

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

  • A36𝑥6𝑥+75𝑥(75𝑥+36𝑥)cotcsccsccotcsc
  • B186𝑥6𝑥+355𝑥(75𝑥+36𝑥)cotcsccsccotcsc
  • C186𝑥6𝑥+355𝑥(75𝑥+36𝑥)cotcsccsccotcsc
  • D186𝑥6𝑥355𝑥(75𝑥+36𝑥)cotcsccsccotcsc

Q9:

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

  • A𝑦=2𝜃(𝜃)(𝜃)coscotsincscsin
  • B𝑦=2𝜃(𝜃)(𝜃)coscotsincscsin
  • C𝑦=2𝜃(𝜃)(𝜃)coscotsincscsin
  • D𝑦=𝜃(𝜃)(𝜃)coscotsincscsin
  • E𝑦=𝜃(𝜃)(𝜃)coscotsincscsin

Q10:

If 𝑦=(𝑥+8𝑥)(𝑥8𝑥)csccotcsccot, find 𝑦.

  • A𝑥𝑥+88𝑥8𝑥coscsccoscsc
  • B2𝑥𝑥+168𝑥8𝑥coscsccoscsc
  • C2𝑥𝑥+28𝑥8𝑥coscsccoscsc
  • D2𝑥𝑥+168𝑥8𝑥coscsccoscsc

Q11:

Given that 𝑦=7𝑥+21𝑥cot, find dd𝑦𝑥.

  • A21𝑥𝑥21𝑥𝑥1𝑥csc
  • B35𝑥𝑥2+1𝑥1𝑥csc
  • C35𝑥𝑥2+1𝑥𝑥1𝑥csc
  • D35𝑥𝑥221𝑥csc

Q12:

Given that 𝑦=73𝑥3𝑥4cot, find dd𝑦𝑥.

  • A(63𝑥84)3𝑥213𝑥(3𝑥4)csccot
  • B(63𝑥84)3𝑥73𝑥(3𝑥4)csccot
  • C(21𝑥28)3𝑥213𝑥(3𝑥4)csccot
  • D(21𝑥28)3𝑥213𝑥(3𝑥4)csccot
  • E(63𝑥84)3𝑥213𝑥(3𝑥4)csccot

Q13:

If 𝑦=85𝑥6sec, find dd𝑦𝑥.

  • A165𝑥5𝑥tansec
  • B165𝑥sec
  • C405𝑥5𝑥tansec
  • D805𝑥5𝑥tansec
  • E805𝑥sec

Q14:

Given that 𝑦=5𝑥4𝑥cot, find dd𝑦𝑥.

  • A20𝑥4𝑥+10𝑥4𝑥csccot
  • B5𝑥4𝑥+10𝑥4𝑥csccot
  • C20𝑥4𝑥+10𝑥4𝑥csccot
  • D20𝑥4𝑥+5𝑥4𝑥csccot
  • E20𝑥4𝑥+5𝑥4𝑥csccot

Q15:

If 𝑦=19𝑥+18csc, find dd𝑦𝑥.

  • A19𝑥𝑥219𝑥+18cotcsccsc
  • B19𝑥𝑥219𝑥+18cotcsccsc
  • C19𝑥219𝑥+18tancsc
  • D19𝑥𝑥219𝑥+18tancsccsc

Q16:

Find dd𝑦𝑥, given that 𝑦=96𝑥7𝑥tancsc.

  • A546𝑥+77𝑥tancsc
  • B77𝑥7𝑥+546𝑥cotcscsec
  • C77𝑥546𝑥cotsec
  • D77𝑥546𝑥cotsec
  • E77𝑥7𝑥546𝑥cotcscsec

Q17:

Given that 𝑦=9𝑥+53𝑥cot, find dd𝑦𝑥.

  • A9𝑥153𝑥3𝑥29𝑥+53𝑥cotcsccot
  • B18𝑥+303𝑥3𝑥29𝑥+53𝑥cotcsccot
  • C18𝑥303𝑥3𝑥29𝑥+53𝑥cotcsccot
  • D18𝑥+103𝑥3𝑥9𝑥+53𝑥cotcsccot

Q18:

Given that 𝑦=3(𝑥+2)csc, find dd𝑦𝑥.

  • A30𝑥(𝑥+2)(𝑥+2)csccot
  • B30𝑥(𝑥+2)(𝑥+2)csccot
  • C10(𝑥+2)(𝑥+2)csccot
  • D6(𝑥+2)(𝑥+2)csccot

Q19:

Given that 𝑦=47(8𝑥)sectan, find dd𝑦𝑥.

  • A87(8𝑥)(8𝑥)tantansectan
  • B47(8𝑥)(8𝑥)(8𝑥)tantansecsectan
  • C647(8𝑥)(8𝑥)(8𝑥)tantansecsectan
  • D47(8𝑥)(8𝑥)tantansectan

Q20:

Differentiate 𝑦=𝑥3𝑥seccsc.

  • A𝑦=𝑥𝑥3𝑥𝑥sectancsccot
  • B𝑦=𝑥𝑥+3𝑥𝑥seccotcsctan
  • C𝑦=𝑥𝑥+3𝑥𝑥sectancsccot
  • D𝑦=𝑥𝑥+3𝑥𝑥sectancsccot
  • E𝑦=𝑥𝑥3𝑥𝑥sectancsccot

Q21:

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

  • A243
  • B12
  • C63
  • D243

Q22:

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

  • A𝑦+11𝑥2332+𝜋6=0
  • B𝑦+11𝑥211𝜋12+332=0
  • C𝑦11𝑥2332+11𝜋12=0
  • D𝑦+11𝑥211𝜋12332=0

Q23:

If 𝑦=98𝑥8𝑥tansec, find dd𝑦𝑥.

  • A728𝑥8𝑥728𝑥tansecsec
  • B98𝑥8𝑥98𝑥tansecsec
  • C98𝑥98𝑥tansec
  • D728𝑥728𝑥tansec

Q24:

Find dd𝑦𝑥, given that 𝑦=(35𝑥+7)cot.

  • A60(35𝑥+7)5𝑥cotcsc
  • B60(35𝑥+7)cot
  • C4(35𝑥+7)5𝑥cotcsc
  • D60(35𝑥+7)5𝑥cotcsc
  • E20(35𝑥+7)cot

Q25:

Given 𝑦=(𝑥+3)(9𝑥+𝑥)csc, find dd𝑦𝑥.

  • A18𝑥+(𝑥+3)𝑥𝑥+𝑥+27cotcsccsc
  • B9𝑥(𝑥+3)𝑥𝑥+𝑥+27cotcsccsc
  • C18𝑥(𝑥+3)𝑥𝑥+𝑥+27cotcsccsc
  • D18𝑥(𝑥3)𝑥𝑥+𝑥+27cotcsccsc

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