Worksheet: Differentiation of Inverse Functions

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

Q1:

Use the graph of 𝑦=𝑓(𝑥) to estimate (𝑓)(2).

  • A3
  • B12
  • C12
  • D2
  • E2

Q2:

Use the inverse function theorem to find the derivative of 𝑔(𝑥)=𝑥+3𝑥.

  • A3𝑥
  • B3𝑥1
  • C3(𝑥1)
  • D𝑥3
  • E𝑥13

Q3:

Consider the function 𝑓(𝑥)=𝑥2𝑥+3𝑥+2 and 𝑃(4,1).

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

  • A12
  • B4
  • C14
  • D1
  • E2

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

  • A𝑦=12𝑥1
  • B𝑦=14𝑥
  • C𝑦=𝑥3
  • D𝑦=2𝑥7
  • E𝑦=4𝑥15

Q4:

A wall is 15 feet high. An angle 𝜃 is formed when a ladder of length 𝑥 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 dd𝜃𝑥 when the ladder has a length of 21 feet.

Q5:

If 𝑓(2𝜋)=1, 𝑓(2𝜋)=1, and 𝑎=1, find 𝑓(𝑎).

Q6:

If 𝑓12=6, 𝑓12=3, and 𝑎=6, find 𝑓(𝑎).

  • A12
  • B3
  • C2
  • D6
  • E13

Q7:

For the function 𝑓(𝑥)=2𝑥+3𝑥sin at 𝑎=0, find 𝑓(𝑎).

  • A5
  • B15
  • C15
  • D5
  • E0

Q8:

For the function 𝑓(𝑥)=2𝑥+6𝑥+10 at 𝑎=2, find 𝑓(𝑎).

  • A14
  • B12
  • C112
  • D2
  • E114

Q9:

For the function 𝑓(𝑥)=3𝑥+2𝑥tan at 𝑎=0, 1<𝑥<1, find 𝑓(𝑎).

  • A3
  • B13
  • C13
  • D3
  • E0

Q10:

For the function 𝑓(𝑥)=𝑥3𝑥 at 𝑎=2, 𝑥>0, find 𝑓(𝑎).

  • A13
  • B2
  • C43
  • D3
  • E34

Q11:

Consider the function 𝑓(𝑥)=𝑥+2𝑥1 and 𝑃(11,2).

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

  • A365
  • B14
  • C114
  • D11
  • E1365

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

  • A𝑦=14𝑥17
  • B𝑦=114𝑥+1714
  • C𝑦=2𝑥20
  • D𝑦=1365𝑥+719365
  • E𝑦=11𝑥119

Q12:

Use the inverse function theorem to find the derivative of 𝑔(𝑥)=𝑥.

  • A𝑥
  • B5𝑥
  • C4𝑥
  • D45𝑥
  • E15𝑥

Q13:

If 𝑓(5)=3, 𝑓(5)=14, and 𝑎=3, find 𝑓(𝑎).

Q14:

For the function 𝑓(𝑥)=3𝑥+𝑥 at 𝑎=4, find 𝑓(𝑎).

  • A72
  • B4
  • C14
  • D114
  • E4

Q15:

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

𝑥𝑓(𝑥)𝑔(𝑥)𝑓(𝑥)
15913
0310

Q16:

Let 𝑔 be the inverse of 𝑓. Using the values in the table, find 𝑔(1).

𝑥12
𝑓(𝑥)59
𝑔(𝑥)24
𝑓(𝑥)21

Q17:

Given that 𝑥=𝑦+𝑦+𝑦, find dd𝑦𝑥.

  • A6𝑦30𝑦4𝑦+3𝑦
  • B6𝑦30𝑦+4𝑦+3𝑦
  • C6𝑦30𝑦+4𝑦+3𝑦
  • D5𝑦+12𝑦+3𝑦𝑦
  • E5𝑦+12𝑦+32𝑦

Q18:

Given that 𝑥=𝑒, find dd𝑦𝑥, giving your answer in terms of 𝑥.

  • Add𝑥𝑦=𝑒
  • Bdd𝑦𝑥=1𝑦
  • Cdd𝑦𝑥=1
  • Ddd𝑦𝑥=𝑥
  • Edd𝑦𝑥=1𝑥

Q19:

Consider the function 𝑦=𝑥, where 𝑥0.

Find dd𝑦𝑥.

  • A0
  • B2𝑥
  • C2𝑥
  • D13𝑥
  • E𝑥

Express 1dd in terms of 𝑦.

  • A12𝑥
  • B1𝑥
  • C23𝑥
  • D12𝑥
  • E12𝑦

By expressing 𝑥 in terms of 𝑦, find an expression for dd𝑥𝑦.

  • A12𝑥
  • B23𝑥
  • C12𝑦
  • D12𝑥
  • E1𝑥

Q20:

Let 𝑓(𝑥)=12𝑥+12𝑥+5𝑥4 and let 𝑔 be the inverse of 𝑓. Given that 𝑓(2)=12, what is 𝑔(12)?

  • A12,796
  • B126
  • C112
  • D113
  • E1223

Q21:

Given that 𝑥=𝑒+𝑦sin, find dd𝑦𝑥.

  • A1𝑒+𝑦cos
  • B𝑒𝑦cos
  • C𝑒+𝑦cos
  • D1𝑒+𝑦cos
  • E1𝑒𝑦cos

Q22:

Let 𝑓(𝑥)=12𝑥+12𝑥+5𝑥4 and let 𝑔 be the inverse of 𝑓. Given that 𝑓(2)=12, what is 𝑔(12)?

  • A19
  • B233
  • C113
  • D1233
  • E13

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