Worksheet: Derivatives of Inverse Hyperbolic Functions

In this worksheet, we will practice finding the derivatives of inverse hyperbolic functions and using rules of differentiation with them.

Q1:

Find the derivative of the function 𝑦 = √ π‘₯ c o s h   .

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

Q2:

Find the derivative of the function 𝑦 = π‘₯ ο€» π‘₯ 3  βˆ’ √ 9 + π‘₯ s i n h    .

  • A s i n h    ο€» π‘₯ 3  + 2 π‘₯ 9 + π‘₯
  • B s i n h     ο€» π‘₯ 3  βˆ’ π‘₯ 9 βˆ’ π‘₯ + π‘₯ 9 + π‘₯
  • C s i n h     ο€» π‘₯ 3  + π‘₯ 3 √ π‘₯ + 1 + π‘₯ √ π‘₯ + 9
  • D s i n h   ο€» π‘₯ 3 
  • E s i n h    ο€» π‘₯ 3  βˆ’ 2 π‘₯ 9 + π‘₯

Q3:

Find the derivative of the function 𝑦 = ( π‘₯ ) s i n h t a n   .

  • A βˆ’ π‘₯ √ 1 βˆ’ π‘₯ s e c t a n  
  • B s e c t a n   π‘₯ √ π‘₯ βˆ’ 1
  • C βˆ’ π‘₯ s e c
  • D | π‘₯ | s e c
  • E s e c t a n   π‘₯ 1 βˆ’ π‘₯

Q4:

Find the derivative of the function 𝑦 = ( π‘₯ ) c o t h s e c   .

  • A c s c π‘₯
  • B βˆ’ π‘₯ s e c
  • C c o t π‘₯
  • D βˆ’ π‘₯ c s c
  • E s e c π‘₯

Q5:

Find the derivative of t a n h   ( π‘₯ ) by using the inverse function theorem together with the facts that d d t a n h c o s h π‘₯ π‘₯ = 1 π‘₯  and c o s h s i n h   π‘₯ βˆ’ π‘₯ = 1 .

  • A c o s h  π‘₯
  • B 1 1 βˆ’ π‘₯ 
  • C s i n h  π‘₯
  • D 1 1 + π‘₯ 
  • E 1 π‘₯ βˆ’ 1 

Q6:

Find the derivative of the function 𝑦 = π‘₯ π‘₯ + √ 1 βˆ’ π‘₯ t a n h l n    .

  • A βˆ’ π‘₯ t a n h  
  • B t a n h    π‘₯ βˆ’ 2 π‘₯ 1 βˆ’ π‘₯
  • C t a n h    π‘₯ + 2 π‘₯ 1 βˆ’ π‘₯
  • D t a n h   π‘₯
  • E t a n h     π‘₯ + 2 π‘₯ √ 1 βˆ’ π‘₯ + π‘₯ 1 βˆ’ π‘₯

Q7:

Find the derivative of the function 𝑦 = 𝑒 s e c h     .

  • A 1 √ 1 + 𝑒   
  • B 1 √ 1 βˆ’ 𝑒   
  • C 𝑒 π‘₯ √ 1 βˆ’ π‘₯   
  • D 𝑒 √ 1 βˆ’ 𝑒     
  • E βˆ’ 1 √ 1 βˆ’ 𝑒   

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