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Worksheet: Higher-Order Implicit Differentiation

Q1:

Given that π‘₯ + 3 𝑦 = 3 2 2 , determine 𝑦 β€² β€² by implicit differentiation.

  • A 𝑦 β€² β€² = βˆ’ 2 π‘₯ + 3 9 𝑦 2 2
  • B 𝑦 β€² β€² = βˆ’ 𝑦 + 1 1 2 𝑦 2 2
  • C 𝑦 β€² β€² = 2 𝑦 βˆ’ 1 3 𝑦 2 3
  • D 𝑦 β€² β€² = βˆ’ 1 3 𝑦 3
  • E 𝑦 β€² β€² = 1 3 𝑦 3

Q2:

Given that π‘₯ + 9 = βˆ’ 2 π‘₯ 𝑦 2 , find π‘₯ 𝑦 π‘₯ + 2 𝑦 π‘₯ d d d d 2 2 .

  • A βˆ’ 1 2
  • B2
  • C βˆ’ 4
  • D βˆ’ 1

Q3:

Given that π‘₯ βˆ’ 5 = 7 π‘₯ 𝑦 2 , find π‘₯ 𝑦 π‘₯ + 2 𝑦 π‘₯ d d d d 2 2 .

  • A 1 7
  • B2
  • C14
  • D 2 7

Q4:

Find d d 𝑦 π‘₯ by implicit differentiation if βˆ’ 𝑒 π‘₯ = 4 π‘₯ 𝑦 + 2 π‘₯ 𝑦 s i n .

  • A 𝑒 π‘₯ + 4 𝑦 + 2 𝑒 π‘₯ + 4 π‘₯ 𝑦 𝑦 c o s s i n
  • B 𝑒 π‘₯ βˆ’ 4 𝑦 βˆ’ 2 𝑒 π‘₯ + 4 π‘₯ 𝑦 𝑦 c o s s i n
  • C βˆ’ 𝑒 π‘₯ + 4 𝑦 𝑒 π‘₯ + 4 π‘₯ 𝑦 𝑦 c o s s i n
  • D βˆ’ 𝑒 π‘₯ + 4 𝑦 + 2 𝑒 π‘₯ + 4 π‘₯ 𝑦 𝑦 c o s s i n
  • E 𝑒 π‘₯ βˆ’ 4 𝑦 𝑒 π‘₯ + 4 π‘₯ 𝑦 𝑦 c o s s i n

Q5:

Given that 2 π‘₯ βˆ’ π‘₯ 𝑦 βˆ’ 𝑦 = βˆ’ 1 2 2 , determine 𝑦 β€² β€² by implicit differentiation.

  • A 𝑦 = 9 π‘₯ 𝑦 β€² βˆ’ 9 𝑦 ( 2 𝑦 + π‘₯ ) β€² β€² 2
  • B 𝑦 = βˆ’ 7 π‘₯ 𝑦 β€² βˆ’ 7 𝑦 ( 2 𝑦 + π‘₯ ) β€² β€² 2
  • C 𝑦 = βˆ’ 9 π‘₯ βˆ’ 9 𝑦 ( 2 𝑦 + π‘₯ ) β€² β€² 2
  • D 𝑦 = βˆ’ 9 π‘₯ 𝑦 β€² βˆ’ 9 𝑦 ( 2 𝑦 + π‘₯ ) β€² β€² 2
  • E 𝑦 = 9 π‘₯ βˆ’ 9 𝑦 ( 2 𝑦 + π‘₯ ) β€² β€² 2

Q6:

Given that π‘₯ + 3 π‘₯ 𝑦 βˆ’ 5 𝑦 = βˆ’ 2 2 2 , determine 𝑦 β€² β€² by implicit differentiation.

  • A 𝑦 = 2 9 π‘₯ 𝑦 β€² βˆ’ 2 9 𝑦 ( βˆ’ 1 0 𝑦 + 3 π‘₯ ) β€² β€² 2
  • B 𝑦 = βˆ’ 1 1 π‘₯ 𝑦 β€² βˆ’ 1 1 𝑦 ( βˆ’ 1 0 𝑦 + 3 π‘₯ ) β€² β€² 2
  • C 𝑦 = βˆ’ 2 9 π‘₯ βˆ’ 2 9 𝑦 ( βˆ’ 1 0 𝑦 + 3 π‘₯ ) β€² β€² 2
  • D 𝑦 = βˆ’ 2 9 π‘₯ 𝑦 β€² βˆ’ 2 9 𝑦 ( βˆ’ 1 0 𝑦 + 3 π‘₯ ) β€² β€² 2
  • E 𝑦 = 2 9 π‘₯ βˆ’ 2 9 𝑦 ( βˆ’ 1 0 𝑦 + 3 π‘₯ ) β€² β€² 2

Q7:

If 𝑒 = 5 π‘₯ βˆ’ 4 𝑦 π‘₯ 𝑦 , determine d d 𝑦 π‘₯ by implicit differentiation.

  • A d d 𝑦 π‘₯ = 𝑦 ο€½ 5 𝑦 βˆ’ 𝑒 + π‘₯ 𝑒  4 𝑦 π‘₯ 𝑦 π‘₯ 𝑦 2
  • B d d 𝑦 π‘₯ = βˆ’ 𝑦 ο€½ 𝑦 βˆ’ 𝑒  π‘₯ 𝑒 π‘₯ 𝑦 π‘₯ 𝑦
  • C d d 𝑦 π‘₯ = 𝑦 ο€½ 5 𝑦 + 𝑒  ο€½ 4 𝑦 + π‘₯ 𝑒  π‘₯ 𝑦 π‘₯ 𝑦 2
  • D d d 𝑦 π‘₯ = 𝑦 ο€½ 5 𝑦 βˆ’ 𝑒  ο€½ 4 𝑦 βˆ’ π‘₯ 𝑒  π‘₯ 𝑦 π‘₯ 𝑦 2
  • E d d 𝑦 π‘₯ = βˆ’ 𝑦 ο€½ 5 𝑦 βˆ’ 𝑒  ο€½ 4 𝑦 βˆ’ π‘₯ 𝑒  π‘₯ 𝑦 π‘₯ 𝑦 2

Q8:

Given that βˆ’ 8 π‘₯ βˆ’ 3 π‘₯ βˆ’ 5 𝑦 = 0 2 2 , find 𝑦 𝑦 π‘₯ + ο€½ 𝑦 π‘₯  d d d d 2 2 2 .

  • A βˆ’ 8
  • B16
  • C 8 5
  • D βˆ’ 8 5

Q9:

If π‘₯ + π‘₯ 𝑦 + 𝑦 = 1 2 3 , find the value of 𝑦 β€² β€² β€² at π‘₯ = 1 .

Q10:

Given that π‘₯ βˆ’ 3 𝑦 = βˆ’ 4 3 3 , find 𝑦 β€² β€² by implicit differentiation.

  • A 𝑦 β€² β€² = βˆ’ π‘₯ + 3 π‘₯ 𝑦 3 𝑦 5 3 2 3
  • B 𝑦 β€² β€² = βˆ’ π‘₯ + π‘₯ 𝑦 3 𝑦 2 3
  • C 𝑦 β€² β€² = βˆ’ π‘₯ + 3 π‘₯ 𝑦 9 𝑦 4 3 4
  • D 𝑦 β€² β€² = βˆ’ 2 π‘₯ + 6 π‘₯ 𝑦 9 𝑦 4 3 5
  • E 𝑦 β€² β€² = βˆ’ 2 π‘₯ + 2 π‘₯ 𝑦 3 𝑦 2 3

Q11:

Find d d 3 3 𝑦 π‘₯ , given that 6 π‘₯ + 6 𝑦 = 2 5 2 2 .

  • A βˆ’ 7 5 π‘₯ 𝑦 5
  • B βˆ’ π‘₯ 2 𝑦 5
  • C βˆ’ 2 5 π‘₯ 2 𝑦 6
  • D βˆ’ 2 5 π‘₯ 2 𝑦 5

Q12:

Let βˆ’ 7 π‘Ž π‘₯ + 5 𝑏 π‘₯ + 𝑦 = 9 𝑐 3 2 , where π‘Ž , 𝑏 , and 𝑐 are constants. Find 𝑦 ο€Ώ 𝑦 π‘₯  + ο€½ 𝑦 π‘₯  βˆ’ 2 1 π‘Ž π‘₯ d d d d 2 2 2 .

Q13:

Given that s i n c o s 𝑦 + 2 π‘₯ = 5 , determine 𝑦 β€² β€² by implicit differentiation.

  • A 𝑦 β€² β€² = 2 π‘₯ 𝑦 βˆ’ 2 π‘₯ 𝑦 𝑦 s i n s i n c o s c o s c o s 2
  • B 𝑦 β€² β€² = βˆ’ 4 π‘₯ 𝑦 + 2 π‘₯ 𝑦 𝑦 s i n s i n c o s c o s c o s 2 2 3
  • C 𝑦 β€² β€² = βˆ’ 4 π‘₯ 𝑦 + 2 π‘₯ 𝑦 𝑦 s i n s i n c o s c o s c o s 2 2 3
  • D 𝑦 β€² β€² = 4 π‘₯ 𝑦 + 2 π‘₯ 𝑦 𝑦 s i n s i n c o s c o s c o s 2 2 3
  • E 𝑦 β€² β€² = 2 π‘₯ 𝑦 + 2 π‘₯ 𝑦 𝑦 s i n s i n c o s c o s c o s 2

Q14:

Given that βˆ’ 7 π‘₯ βˆ’ 7 𝑦 = 1 0 2 2 , determine 𝑦 𝑦 βˆ’ 1 0 7 3 β€² β€² .

Q15:

If 2 π‘₯ 𝑦 = βˆ’ 1 7 5 π‘₯ 5 π‘₯ s i n c o s , find π‘₯ ο€Ώ 𝑦 π‘₯  + 2 ο€½ 𝑦 π‘₯  d d d d 2 2 .

  • A βˆ’ 1 0 0 π‘₯ 𝑦
  • B βˆ’ 2 5 π‘₯ 𝑦
  • C βˆ’ 5 0 π‘₯ 𝑦
  • D βˆ’ 1 0 0 π‘₯ 𝑦

Q16:

If 𝑓 β€² ( π‘₯ ) = π‘₯ 𝑓 ( π‘₯ ) , and 𝑓 ( βˆ’ 9 ) = 6 , then determine 𝑓 β€² β€² ( βˆ’ 9 ) .

Q17:

Given that 9 π‘₯ 𝑒 + 𝑦 𝑒 = 7 βˆ’ βˆ’ 5 𝑦 7 8 π‘₯ 5 , find d d 𝑦 π‘₯ when π‘₯ = 0 .

  • A 5 6 𝑒 βˆ’ 4 5 𝑒 5 5
  • B 5 6 𝑒 βˆ’ 4 5 5 5
  • C 𝑒 βˆ’ 4 5 5 𝑒 5 5
  • D 5 6 𝑒 βˆ’ 4 5 5 𝑒 5 5

Q18:

Given that ( 6 π‘₯ + 7 𝑦 ) = 4 7 , find d d d d 2 2 𝑦 π‘₯ + 𝑦 π‘₯ .

  • A0
  • B 6 7
  • C βˆ’ 2 7
  • D βˆ’ 6 7

Q19:

Given 7 𝑒 π‘₯ = 8 𝑦 + 5 π‘₯ βˆ’ 4 π‘₯ 𝑦 2 s i n , find d d 𝑦 π‘₯ at π‘₯ = 0 .

  • A βˆ’ 7 2
  • B βˆ’ 1 8
  • C0
  • D 7 8

Q20:

If βˆ’ 1 0 π‘₯ 𝑦 βˆ’ 5 = π‘₯ 2 , find π‘₯ ο€Ώ 𝑦 π‘₯  + 2 ο€½ 𝑦 π‘₯  d d d d 2 2 .

  • A βˆ’ 2 5 + 𝑦 π‘₯
  • B 2 + 𝑦 βˆ’ 1 5 π‘₯
  • C βˆ’ 2 5 βˆ’ 𝑦 π‘₯
  • D βˆ’ 1 5
  • E 1 5

Q21:

Suppose that 𝑒 βˆ’ 2 π‘₯ 𝑦 = 𝑒 2 𝑦 3 . Find 𝑦 β€² β€² when π‘₯ = 0 .

  • A 𝑦 β€² β€² = βˆ’ 3 𝑒 4 βˆ’ 3 𝑒 3 6
  • B 𝑦 β€² β€² = βˆ’ 9 2 𝑒 6
  • C 𝑦 β€² β€² = βˆ’ 6 𝑒 + 3 2 𝑒 3 6
  • D 𝑦 β€² β€² = βˆ’ 3 2 𝑒 6
  • E 𝑦 β€² β€² = 3 𝑒 4 βˆ’ 3 𝑒 3 6

Q22:

Given that 5 𝑦 = 9 π‘₯ 3 3 , find 𝑦 𝑦 β€² β€² + 2 ( 𝑦 β€² ) 2 .

  • A 9 π‘₯ 5 𝑦 2
  • B 3 π‘₯ 5 𝑦
  • C 1 8 π‘₯ 5 𝑦 2
  • D 1 8 π‘₯ 5 𝑦

Q23:

Given that βˆ’ 2 𝑦 = π‘₯ 9 7 , find 𝑦 𝑦 β€² β€² + 8 ( 𝑦 β€² ) 2 .

  • A βˆ’ 7 π‘₯ 1 8 𝑦 6 7
  • B βˆ’ π‘₯ 1 8 𝑦 5 7
  • C βˆ’ 7 π‘₯ 3 𝑦 6 7
  • D βˆ’ 7 π‘₯ 3 𝑦 5 7

Q24:

Given that 𝑒 𝑦 + 3 𝑒 π‘₯ = 5 π‘₯ 5 𝑦 5 , determine d d 𝑦 π‘₯ at π‘₯ = 0 .

  • A 3 𝑒 + 1
  • B βˆ’ 3 𝑒 βˆ’ 5
  • C 1 5 𝑒 + 5
  • D βˆ’ 3 𝑒 βˆ’ 1

Q25:

If 8 π‘₯ 𝑦 = 7 4 π‘₯ s i n , determine π‘₯ 𝑦 π‘₯ + 2 𝑦 π‘₯ d d d d 2 2 .

  • A βˆ’ 7 2 4 π‘₯ s i n
  • B βˆ’ 7 8 4 π‘₯ s i n
  • C 7 2 4 π‘₯ s i n
  • D βˆ’ 1 4 4 π‘₯ s i n
  • E 1 4 4 π‘₯ s i n