Worksheet: Derivatives of Hyperbolic Functions

In this worksheet, we will practice differentiating hyperbolic functions and using the rules of differentiation like chain rule and product rule with them.

Q1:

Determine and simplify d d s i n h π‘₯ π‘₯ , where s i n h π‘₯ = 𝑒 βˆ’ 𝑒 2    and c o s h π‘₯ = 𝑒 + 𝑒 2    .

  • A βˆ’ π‘₯ s i n h
  • B c o s h π‘₯
  • C s i n h π‘₯
  • D t a n h π‘₯
  • E βˆ’ π‘₯ c o s h

Q2:

Determine and simplify d d c o s h x π‘₯ , where s i n h π‘₯ = 𝑒 βˆ’ 𝑒 2    and c o s h π‘₯ = 𝑒 + 𝑒 2    .

  • A c o s h π‘₯
  • B βˆ’ π‘₯ s i n h
  • C βˆ’ π‘₯ c o s h
  • D s i n h π‘₯
  • E t a n h π‘₯

Q3:

Find the derivative of t a n h π‘₯ , giving your answer in terms of hyperbolic functions.

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

Q4:

Find the derivative of the function 𝑔 ( 𝑑 ) = 𝑑 √ 𝑑 + 1 c o t h  .

  • A c o t h s e c h √ 𝑑 + 1 βˆ’ 2 𝑑 √ 𝑑 + 1 √ 𝑑 + 1     
  • B c o t h c s c h √ 𝑑 + 1 βˆ’ 2 𝑑 √ 𝑑 + 1 √ 𝑑 + 1     
  • C c o t h c s c h c o t h √ 𝑑 + 1 βˆ’ 𝑑 √ 𝑑 + 1 √ 𝑑 + 1 √ 𝑑 + 1     
  • D c o t h s e c h √ 𝑑 + 1 + 𝑑 √ 𝑑 + 1 √ 𝑑 + 1     
  • E c o t h c s c h √ 𝑑 + 1 βˆ’ 𝑑 √ 𝑑 + 1 √ 𝑑 + 1     

Q5:

Find the derivative of the function β„Ž ( π‘₯ ) = π‘₯ s i n h  .

  • A βˆ’ π‘₯ π‘₯ c o s h 
  • B c o s h π‘₯ 
  • C 2 π‘₯ π‘₯   c o s h
  • D βˆ’ 2 π‘₯ π‘₯ c o s h 
  • E 2 π‘₯ π‘₯ c o s h 

Q6:

Find the derivative of the function 𝑓 ( 𝑑 ) = 1 + 𝑑 1 βˆ’ 𝑑 s i n h s i n h .

  • A 2 𝑑 𝑑 ( 1 βˆ’ 𝑑 ) s i n h c o s h s i n h 
  • B βˆ’ 2 𝑑 1 βˆ’ 𝑑 c o s h s i n h 
  • C 2 𝑑 ( 1 βˆ’ 𝑑 ) c o s h s i n h 
  • D 2 𝑑 𝑑 1 βˆ’ 𝑑 s i n h c o s h s i n h
  • E βˆ’ 2 𝑑 𝑑 ( 1 βˆ’ 𝑑 ) s i n h c o s h s i n h 

Q7:

Find the derivative of the function 𝑔 ( π‘₯ ) = π‘₯ s i n h  .

  • A s i n h 2 π‘₯ 2
  • B βˆ’ π‘₯ 2 s i n h
  • C s i n h π‘₯
  • D s i n h 2 π‘₯
  • E 2 π‘₯ π‘₯ c o s h 

Q8:

Find the derivative of the function 𝑓 ( π‘₯ ) = √ π‘₯ t a n h .

  • A s e c h  √ π‘₯ 2 √ π‘₯
  • B s e c h  √ π‘₯ √ π‘₯
  • C βˆ’ √ π‘₯ 2 √ π‘₯ s e c h 
  • D s e c h √ π‘₯ √ π‘₯
  • E βˆ’ √ π‘₯ 2 √ π‘₯ s e c h

Q9:

Find the derivative of the function 𝑦 = π‘₯ ( 1 + π‘₯ ) s e c h l n s e c h .

  • A βˆ’ π‘₯ π‘₯ ( 2 + π‘₯ ) s e c h t a n h l n s e c h
  • B βˆ’ π‘₯ π‘₯ ( 1 + π‘₯ ) s e c h t a n h l n s e c h
  • C s e c h t a n h l n s e c h π‘₯ π‘₯ ( 1 βˆ’ π‘₯ )
  • D s e c h t a n h l n s e c h π‘₯ π‘₯ ( 2 + π‘₯ )
  • E βˆ’ π‘₯ π‘₯ ( 1 βˆ’ π‘₯ ) s e c h t a n h l n s e c h

Q10:

Find the derivative of the function 𝑦 = 𝑒 c o s h   .

  • A 𝑒 3 π‘₯ c o s h   s i n h
  • B 3 𝑒 3 π‘₯ c o s h   s i n h
  • C 3 𝑒 3 π‘₯ c o s h   c o s h
  • D 𝑒    s i n h
  • E βˆ’ 3 𝑒 3 π‘₯ c o s h   c o s h

Q11:

Find the derivative of the function 𝑓 ( π‘₯ ) = 𝑒 π‘₯  c o s h .

  • A 2 𝑒  
  • B 𝑒 2  
  • C π‘₯ ο€Ί 𝑒 βˆ’ 𝑒      
  • D 2 π‘₯ 𝑒  
  • E 𝑒  

Q12:

Find the derivative of the function 𝐺 ( 𝑑 ) = ( 𝑑 ) s i n h l n .

  • A 𝑑 + 1 𝑑 
  • B 𝑑 + 1 2 𝑑  
  • C0
  • D1
  • E 𝑑 βˆ’ 1 2 𝑑  

Q13:

Find the derivative of the function 𝐹 ( 𝑑 ) = ( 𝑑 ) l n s i n h .

  • A c o t h 𝑑
  • B βˆ’ 𝑑 c o t h
  • C βˆ’ 𝑑 t a n h
  • D c s c h 𝑑
  • E t a n h 𝑑

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