Worksheet: Differentiation of Inverse Functions

In this worksheet, we will practice finding the derivatives of inverse functions.

Q1:

Use the graph of 𝑦 = 𝑓 ( π‘₯ ) to estimate ( 𝑓 ) β€² ( 2 )   .

  • A βˆ’ 3
  • B βˆ’ 1 2
  • C 1 2
  • D βˆ’ 2
  • E2

Q2:

Use the inverse function theorem to find the derivative of 𝑔 ( π‘₯ ) = π‘₯ + 3 π‘₯ .

  • A βˆ’ 3 π‘₯ 
  • B 3 π‘₯ βˆ’ 1
  • C βˆ’ 3 ( π‘₯ βˆ’ 1 ) 
  • D π‘₯ βˆ’ 1 3
  • E π‘₯ 3 

Q3:

Consider the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 2 π‘₯ + 3 π‘₯ + 2   and 𝑃 ( 4 , 1 ) .

Find the slope of the tangent line to its inverse function 𝑓   at the indicated point 𝑃 .

  • A1
  • B 1 2
  • C 1 4
  • D2
  • E4

Find the equation of the tangent line to the graph of 𝑓   at the indicated point 𝑃 .

  • A 𝑦 = 1 4 π‘₯
  • B 𝑦 = 2 π‘₯ βˆ’ 7
  • C 𝑦 = π‘₯ βˆ’ 3
  • D 𝑦 = 1 2 π‘₯ βˆ’ 1
  • E 𝑦 = 4 π‘₯ βˆ’ 1 5

Q4:

A wall is 15 feet high. An angle πœƒ is formed when a ladder of length π‘₯ is placed against the wall such that the top of the ladder is at the top of the wall, as shown in the figure. The ladder's length can be changed, but the top of the ladder always remains at the top of the wall. Find the rate of change, to the nearest ten thousandth, of the angle d d πœƒ π‘₯ when the ladder has a length of 21 feet.

Q5:

If 𝑓 ( 2 πœ‹ ) = βˆ’ 1 , 𝑓 ( 2 πœ‹ ) = 1  , and π‘Ž = βˆ’ 1 , find ο€Ή 𝑓  ( π‘Ž )    .

Q6:

If 𝑓 ο€Ό 1 2  = βˆ’ 6 , 𝑓 ο€Ό 1 2  = 3  , and π‘Ž = βˆ’ 6 , find ο€Ή 𝑓  ( π‘Ž )    .

  • A2
  • B βˆ’ 6
  • C 1 2
  • D3
  • E 1 3

Q7:

For the function 𝑓 ( π‘₯ ) = 2 π‘₯ + 3 π‘₯ s i n at π‘Ž = 0 , find ο€Ή 𝑓  ( π‘Ž )    .

  • A5
  • B βˆ’ 1 5
  • C 1 5
  • D0
  • E βˆ’ 5

Q8:

For the function 𝑓 ( π‘₯ ) = 2 π‘₯ + 6 π‘₯ + 1 0  at π‘Ž = 2 , find ο€Ή 𝑓  ( π‘Ž )    .

  • A12
  • B2
  • C 1 1 2
  • D 1 1 4
  • E14

Q9:

For the function 𝑓 ( π‘₯ ) = 3 π‘₯ + 2 π‘₯ t a n οŠͺ at π‘Ž = 0 , βˆ’ 1 < π‘₯ < 1 , find ο€Ή 𝑓  ( π‘Ž )    .

  • A0
  • B βˆ’ 3
  • C3
  • D 1 3
  • E βˆ’ 1 3

Q10:

For the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 3 π‘₯ at π‘Ž = 2 , π‘₯ > 0 , find ο€Ή 𝑓  ( π‘Ž )    .

  • A 4 3
  • B 3 4
  • C2
  • D 1 3
  • E3

Q11:

Consider the function 𝑓 ( π‘₯ ) = π‘₯ + 2 π‘₯ βˆ’ 1  and 𝑃 ( 1 1 , 2 ) .

Find the slope of the tangent line to its inverse function 𝑓   at the indicated point 𝑃 .

  • A 1 3 6 5
  • B11
  • C 1 1 4
  • D14
  • E365

Find the equation of the tangent line to the graph of 𝑓   at the indicated point 𝑃 .

  • A 𝑦 = 1 1 π‘₯ βˆ’ 1 1 9
  • B 𝑦 = 1 3 6 5 π‘₯ + 7 1 9 3 6 5
  • C 𝑦 = 1 1 4 π‘₯ + 1 7 1 4
  • D 𝑦 = 1 4 π‘₯ βˆ’ 1 7
  • E 𝑦 = 2 π‘₯ βˆ’ 2 0

Q12:

Use the inverse function theorem to find the derivative of 𝑔 ( π‘₯ ) = √ π‘₯  .

  • A βˆ’ 4 5 π‘₯  
  • B 5 π‘₯ οŠͺ
  • C 1 5 π‘₯   
  • D π‘₯ 
  • E 4 π‘₯ 

Q13:

If 𝑓 ( 5 ) = 3 , 𝑓 ( 5 ) = 1 4  , and π‘Ž = 3 , find ο€Ή 𝑓  ( π‘Ž )    .

Q14:

For the function 𝑓 ( π‘₯ ) = 3 π‘₯ + π‘₯ οŠͺ  at π‘Ž = 4 , find ο€Ή 𝑓  ( π‘Ž )    .

  • A4
  • B 1 1 4
  • C14
  • D 7 2
  • E βˆ’ 4

Q15:

Let 𝑔 be the inverse of 𝑓 . Using the values in the table, find 𝑔 β€² ( 0 ) .

π‘₯ 𝑓 ( π‘₯ ) 𝑔 ( π‘₯ ) 𝑓 β€² ( π‘₯ )
βˆ’ 1 5 9 1 3
0 3 βˆ’ 1 0

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