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 ddsinhπ‘₯π‘₯, where sinhπ‘₯=π‘’βˆ’π‘’2ο—οŠ±ο— and coshπ‘₯=𝑒+𝑒2ο—οŠ±ο—.

  • Asinhπ‘₯
  • Bcoshπ‘₯
  • Cβˆ’π‘₯sinh
  • Dβˆ’π‘₯cosh
  • Etanhπ‘₯

Q2:

Determine and simplify ddcoshxπ‘₯, where sinhπ‘₯=π‘’βˆ’π‘’2ο—οŠ±ο— and coshπ‘₯=𝑒+𝑒2ο—οŠ±ο—.

  • Aβˆ’π‘₯cosh
  • Bsinhπ‘₯
  • Ccoshπ‘₯
  • Dβˆ’π‘₯sinh
  • Etanhπ‘₯

Q3:

Find the derivative of tanhπ‘₯, giving your answer in terms of hyperbolic functions.

  • Asechπ‘₯
  • Bβˆ’π‘₯sech
  • Cβˆ’π‘₯sech
  • Dsechπ‘₯
  • Ecothπ‘₯

Q4:

Find the derivative of the function 𝑔(𝑑)=π‘‘βˆšπ‘‘+1coth.

  • Acothcschβˆšπ‘‘+1βˆ’2π‘‘βˆšπ‘‘+1βˆšπ‘‘+1
  • Bcothcschβˆšπ‘‘+1βˆ’π‘‘βˆšπ‘‘+1βˆšπ‘‘+1
  • Ccothsechβˆšπ‘‘+1βˆ’2π‘‘βˆšπ‘‘+1βˆšπ‘‘+1
  • Dcothsechβˆšπ‘‘+1+π‘‘βˆšπ‘‘+1βˆšπ‘‘+1
  • Ecothcschcothβˆšπ‘‘+1βˆ’π‘‘βˆšπ‘‘+1βˆšπ‘‘+1βˆšπ‘‘+1

Q5:

Find the derivative of the function β„Ž(π‘₯)=π‘₯sinh.

  • A2π‘₯π‘₯cosh
  • Bcoshπ‘₯
  • Cβˆ’2π‘₯π‘₯cosh
  • Dβˆ’π‘₯π‘₯cosh
  • E2π‘₯π‘₯cosh

Q6:

Find the derivative of the function 𝑓(𝑑)=1+𝑑1βˆ’π‘‘sinhsinh.

  • A2𝑑𝑑(1βˆ’π‘‘)sinhcoshsinh
  • B2𝑑(1βˆ’π‘‘)coshsinh
  • C2𝑑𝑑1βˆ’π‘‘sinhcoshsinh
  • Dβˆ’2𝑑𝑑(1βˆ’π‘‘)sinhcoshsinh
  • Eβˆ’2𝑑1βˆ’π‘‘coshsinh

Q7:

Find the derivative of the function 𝑔(π‘₯)=π‘₯sinh.

  • Asinh2π‘₯
  • Bβˆ’π‘₯2sinh
  • Csinhπ‘₯
  • D2π‘₯π‘₯cosh
  • Esinh2π‘₯2

Q8:

Find the derivative of the function 𝑓(π‘₯)=√π‘₯tanh.

  • Asech√π‘₯2√π‘₯
  • Bβˆ’βˆšπ‘₯2√π‘₯sech
  • Cβˆ’βˆšπ‘₯2√π‘₯sech
  • Dsech√π‘₯√π‘₯
  • Esech√π‘₯√π‘₯

Q9:

Find the derivative of the function 𝑦=π‘₯(1+π‘₯)sechlnsech.

  • Asechtanhlnsechπ‘₯π‘₯(1βˆ’π‘₯)
  • Bβˆ’π‘₯π‘₯(2+π‘₯)sechtanhlnsech
  • Cβˆ’π‘₯π‘₯(1βˆ’π‘₯)sechtanhlnsech
  • Dsechtanhlnsechπ‘₯π‘₯(2+π‘₯)
  • Eβˆ’π‘₯π‘₯(1+π‘₯)sechtanhlnsech

Q10:

Find the derivative of the function 𝑦=𝑒coshοŠ©ο—.

  • A𝑒3π‘₯coshοŠ©ο—sinh
  • B3𝑒3π‘₯coshοŠ©ο—cosh
  • C3𝑒3π‘₯coshοŠ©ο—sinh
  • Dβˆ’3𝑒3π‘₯coshοŠ©ο—cosh
  • Eπ‘’οŠ©οŠ©ο—sinh

Q11:

Find the derivative of the function 𝑓(π‘₯)=𝑒π‘₯cosh.

  • Aπ‘₯ο€Ίπ‘’βˆ’π‘’ο†ο—οŠ±ο—οŽ‘οŽ‘
  • Bπ‘’οŠ¨ο—
  • C2π‘₯π‘’ο—οŽ‘
  • D2π‘’οŠ¨ο—
  • E𝑒2οŠ¨ο—

Q12:

Find the derivative of the function 𝐺(𝑑)=(𝑑)sinhln.

  • A𝑑+12π‘‘οŠ¨οŠ¨
  • B0
  • C1
  • Dπ‘‘βˆ’12π‘‘οŠ¨οŠ¨
  • E𝑑+1π‘‘οŠ¨

Q13:

Find the derivative of the function 𝐹(𝑑)=(𝑑)lnsinh.

  • Aβˆ’π‘‘coth
  • Btanh𝑑
  • Cβˆ’π‘‘tanh
  • Dcoth𝑑
  • Ecsch𝑑

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