Worksheet: The Chain Rule

In this worksheet, we will practice finding the derivatives of composite functions using the chain rule.

Q1:

Find the first derivative of the function 𝑦=5𝑥6.

  • A 6 0 𝑥 5 𝑥 6
  • B 6 5 𝑥 6
  • C 6 0 𝑥 5 𝑥 6
  • D 6 5 𝑥 6

Q2:

If 𝑦=8𝑥cos, find dd𝑦𝑥.

  • A 4 0 𝑥 8 𝑥 c o s
  • B 4 0 𝑥 8 𝑥 c o s
  • C 8 𝑥 8 𝑥 c o s
  • D 4 0 𝑥 8 𝑥 s i n
  • E 8 𝑥 8 𝑥 s i n

Q3:

Determine the derivative of 𝑦=2𝑥3𝑥+4.

  • A 𝑦 = 4 𝑥 3 𝑥 2 𝑥 3 𝑥 + 4
  • B 𝑦 = ( 4 𝑥 3 ) 2 𝑥 3 𝑥 + 4
  • C 𝑦 = 5 5 ( 4 𝑥 3 ) 2 𝑥 3 𝑥 + 4
  • D 𝑦 = 5 5 ( 4 𝑥 3 ) 2 𝑥 3 𝑥 + 4
  • E 𝑦 = 5 5 4 𝑥 3 𝑥 2 𝑥 3 𝑥 + 4

Q4:

Find dd𝑦𝑥 if 𝑦=(5𝑥)tancot.

  • A 5 𝑥 ( 5 𝑥 ) c s c s e c c o t
  • B 5 𝑥 ( 5 𝑥 ) c s c s e c c o t
  • C 5 𝑥 ( 5 𝑥 ) c s c s e c c o t
  • D 5 𝑥 ( 5 𝑥 ) c s c s e c c o t
  • E 5 𝑥 ( 5 𝑥 ) c s c s e c c o t

Q5:

Find the first derivative of 𝑦𝑦=73𝑥sectan.

  • A 2 1 𝑥 3 𝑥 𝑥 3 𝑥 t a n t a n t a n s e c s e c t a n
  • B 6 3 𝑥 3 𝑥 𝑥 3 𝑥 t a n t a n t a n s e c s e c t a n
  • C 6 3 𝑥 3 𝑥 𝑥 3 𝑥 t a n t a n t a n s e c s e c t a n
  • D 6 3 𝑥 3 𝑥 𝑥 3 𝑥 t a n t a n t a n s e c s e c t a n

Q6:

Find the first derivative of the function 𝑦=8𝑥9𝑥sin.

  • A 4 4 9 𝑥 9 𝑥 8 𝑥 9 𝑥 s i n c o s s i n
  • B 4 3 6 9 𝑥 8 𝑥 9 𝑥 s i n s i n
  • C 4 + 3 6 9 𝑥 9 𝑥 8 𝑥 9 𝑥 s i n c o s s i n
  • D 4 3 6 9 𝑥 9 𝑥 8 𝑥 9 𝑥 s i n c o s s i n

Q7:

If 𝑦=8𝑥5𝑥sin, find dd𝑦𝑥.

  • A 8 𝑥 5 𝑥 2 0 5 𝑥 5 𝑥 + 8 s i n s i n c o s
  • B 2 0 5 𝑥 5 𝑥 + 8 s i n c o s
  • C 1 0 5 𝑥 5 𝑥 + 4 8 𝑥 5 𝑥 s i n c o s s i n
  • D 1 0 5 𝑥 5 𝑥 + 4 8 𝑥 5 𝑥 s i n c o s s i n
  • E 2 0 5 𝑥 5 𝑥 + 8 8 𝑥 5 𝑥 s i n c o s s i n

Q8:

If 𝑦=𝑥9𝑥+5cos, find dd𝑦𝑥.

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

Q9:

Find the first derivative of the function 𝑦=44𝑥sincsc.

  • A 3 2 𝑥 4 𝑥 4 𝑥 4 4 𝑥 c s c c o t c o s c s c
  • B 3 2 𝑥 4 𝑥 4 𝑥 4 4 𝑥 c s c t a n c o s c s c
  • C 3 2 𝑥 4 𝑥 4 4 𝑥 c o t c o s c s c
  • D 3 2 𝑥 4 𝑥 4 𝑥 4 4 𝑥 c s c c o t c o s c s c

Q10:

If 𝑦=8(6𝑥)(6𝑥)sinsincossin, find dd𝑦𝑥.

  • A 6 6 𝑥 8 ( 6 𝑥 ) + ( 6 𝑥 ) c o s c o s s i n s i n s i n
  • B 6 6 𝑥 8 ( 6 𝑥 ) + ( 6 𝑥 ) c o s c o s s i n s i n s i n
  • C c o s c o s s i n c o s s i n 6 𝑥 8 ( 6 𝑥 ) + ( 6 𝑥 )
  • D 6 6 𝑥 8 ( 6 𝑥 ) ( 6 𝑥 ) c o s c o s s i n c o s s i n

Q11:

Find the derivative of the function 𝑦=(𝜋𝑥)cossintan.

  • A 𝑦 = 𝜋 ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) 2 ( 𝜋 𝑥 ) c o s t a n s i n s i n t a n s i n t a n
  • B 𝑦 = 2 𝜋 ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) c o s t a n s e c s i n s i n t a n s i n t a n
  • C 𝑦 = 𝜋 ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) 2 ( 𝜋 𝑥 ) c o s t a n s e c s i n s i n t a n s i n t a n
  • D 𝑦 = 𝜋 ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) 2 ( 𝜋 𝑥 ) c o s t a n s e c s i n s i n t a n s i n t a n
  • E 𝑦 = 𝜋 ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) ( 𝜋 𝑥 ) c o s t a n s e c s i n s i n t a n s i n t a n

Q12:

Find the first derivative of the function 𝑦=2(4𝑥)cotcos.

  • A 2 4 𝑥 ( 4 𝑥 ) s i n c s c c o s
  • B 8 4 𝑥 ( 4 𝑥 ) s i n c s c c o s
  • C 2 4 𝑥 ( 4 𝑥 ) s i n c s c c o s
  • D 8 4 𝑥 ( 4 𝑥 ) s i n c s c c o s

Q13:

If 𝑦=𝑥sincos, find dd𝑦𝑥.

  • A 3 𝑥 𝑥 𝑥 s i n c o s c o s c o s
  • B 3 𝑥 𝑥 s i n c o s
  • C 3 𝑥 𝑥 𝑥 s i n c o s c o s c o s
  • D 3 𝑥 𝑥 s i n c o s
  • E 3 𝑥 𝑥 c o s c o s c o s

Q14:

If 𝑦=(8𝑥4)sin, find dd𝑦𝑥.

  • A ( 8 𝑥 4 ) c o s
  • B 1 6 𝑥 ( 8 𝑥 4 ) c o s
  • C 1 6 𝑥 ( 8 𝑥 4 ) s i n
  • D 1 6 𝑥 ( 8 𝑥 4 ) c o s

Q15:

If 𝑦=66𝑥11tan, find dd𝑦𝑥.

  • A 7 2 𝑥 6 𝑥 1 1 s e c
  • B 7 2 𝑥 6 𝑥 1 1 s e c
  • C 7 2 𝑥 6 𝑥 1 1 s e c
  • D 7 2 6 𝑥 1 1 s e c
  • E 6 6 𝑥 1 1 s e c

Q16:

If 𝑦=52𝑥cos, find dd𝑦𝑥.

  • A 3 0 2 𝑥 2 𝑥 c o s s i n
  • B 3 0 2 𝑥 2 𝑥 c o s s i n
  • C 3 0 2 𝑥 s i n
  • D 5 2 𝑥 2 𝑥 c o s s i n
  • E 3 0 2 𝑥 c o s

Q17:

If 𝑦=(125𝑥)coscos, find dd𝑦𝑥.

  • A 1 2 ( 1 2 5 𝑥 ) 5 𝑥 s i n c o s s i n
  • B ( 1 2 5 𝑥 ) s i n c o s
  • C 6 0 ( 1 2 5 𝑥 ) 5 𝑥 s i n c o s s i n
  • D 1 2 ( 1 2 5 𝑥 ) 5 𝑥 s i n c o s s i n
  • E 6 0 ( 1 2 5 𝑥 ) 5 𝑥 s i n c o s s i n

Q18:

Find the first derivative of the function 𝑦=6𝑥cot.

  • A c s c 6 𝑥
  • B 4 8 𝑥 6 𝑥 c s c
  • C 4 8 𝑥 6 𝑥 c s c
  • D 4 8 𝑥 6 𝑥 c s c

Q19:

If 𝑦=2+4𝑥tan, find dd𝑦𝑥.

  • A 2 2 4 3 2 + 4 𝑥 𝑥 𝑥 t a n t a n s e c
  • B 2 2 4 3 2 + 4 𝑥 𝑥 𝑥 t a n t a n s e c
  • C 2 2 4 3 2 + 4 𝑥 𝑥 𝑥 t a n t a n s e c
  • D 7 3 2 + 4 𝑥 𝑥 𝑥 t a n t a n s e c

Q20:

If 𝑦=(27𝑥+27𝑥)sincos, find dd𝑦𝑥.

  • A 5 6 ( 7 𝑥 + 7 𝑥 ) s i n c o s
  • B 4 7 𝑥 + 4 7 𝑥 s i n c o s
  • C 8 1 4 𝑥 c o s
  • D 5 6 1 4 𝑥 c o s

Q21:

If 𝑦=17𝑥sin, find dd𝑦𝑥.

  • A 1 7 𝑥 1 7 𝑥 c o s
  • B c o s 1 7 𝑥
  • C 1 7 𝑥 1 7 𝑥 s i n
  • D 1 5 3 𝑥 1 7 𝑥 c o s

Q22:

If 𝑦=9𝑥tan, find dd𝑦𝑥.

  • A 9 𝑥 2 𝑥 s e c
  • B 9 𝑥 s e c
  • C 9 𝑥 2 𝑥 s e c
  • D 9 𝑥 2 𝑥 s e c

Q23:

If 𝑦=810𝑥cos, find dd𝑦𝑥.

  • A 8 1 0 𝑥 1 0 𝑥 s i n c o s
  • B 8 0 1 0 𝑥 1 0 𝑥 s i n c o s
  • C 4 0 1 0 𝑥 1 0 𝑥 s i n c o s
  • D 4 0 1 0 𝑥 c o s
  • E 8 0 1 0 𝑥 1 0 𝑥 s i n c o s

Q24:

If 𝑦=9𝑥tan, find dd𝑦𝑥.

  • A 8 1 9 𝑥 9 𝑥 t a n s e c
  • B 8 1 9 𝑥 9 𝑥 t a n s e c
  • C 8 1 9 𝑥 9 𝑥 t a n s e c
  • D 8 1 9 𝑥 9 𝑥 t a n s e c
  • E 9 9 𝑥 9 𝑥 t a n s e c

Q25:

If 𝑦=6𝑥7+9𝑥tan, find dd𝑦𝑥.

  • A 8 6 𝑥 7 + 9 𝑥 6 𝑥 7 + 9 𝑥 t a n s e c
  • B 2 4 2 + 9 𝑥 6 𝑥 7 + 9 𝑥 s e c
  • C 2 4 2 + 9 𝑥 6 𝑥 7 + 9 𝑥 6 𝑥 7 + 9 𝑥 t a n s e c
  • D 2 4 2 + 9 𝑥 6 𝑥 7 + 9 𝑥 t a n
  • E 2 4 2 + 9 𝑥 6 𝑥 7 + 9 𝑥 6 𝑥 7 + 9 𝑥 t a n s e c

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