# Edexcel A Level Further Maths: Core Pure:复习笔记4.1.4 Differentiating & Integrating Hyperbolic Functions

### Differentiating Hyperbolic Functions

#### What are the derivatives of the hyperbolic functions?

• These are given in the formulae booklet
• You can prove them by differentiating the definitions involving e
• Notice that they are similar to the derivatives of the circular trig functions
• Be careful of the difference between the derivatives of cosx and coshx
• One involves a negative sign and the other does not

#### How do I differentiate expressions involving hyperbolic functions?

• The following differentiation skills may be required
• Chain rule
• Product rule
• Quotient rule
• Implicit differentiation
• Questions may involve showing or proving given results or finding unknown constants
• It is common that derivatives can be written in terms of the original function
• This is due to the derivative of ex also being ex giving rise to the repetition of terms

#### What are the derivatives of the inverse hyperbolic functions?

• These are given in the formulae booklet

#### How do I prove or show the derivatives of the inverse hyperbolic functions?

• Use the same method for differentiating any inverse function
• STEP 1
Write x in terms of y

• can be written
• STEP 2
Differentiate with respect to y

• STEP 3
Write the derivative in terms of x

• STEP 4
Take the reciprocal

• STEP 5
Use the graph to determine whether it is positive of negative

• The graph of  has a positive gradient everywhere

#### Exam Tip

• It is usually easier to differentiate hyperbolic functions using the “trig style” standard results but if you are stuck you can try using their exponential form from the definitions

#### Worked Example

a) Given that , show that .

b) Given that , show that  where  and  are constants to be found.

### Integrating Hyperbolic Functions

#### What are the integrals of the hyperbolic functions?

• These are the reverse results of the derivatives, remembering “+c” of course!
• These are given in the integration section of the formulae booklet
• This can be deduced from the differentiation section of the formulae booklet
• There is also the integral of tanhx
• This is given in the integration section of the formulae booklet
• It can be shown using the substitution
• Since cosh  ≥ 1 for all values of x so there is no need for the modulus signs that usually accompany integrals involving ln

#### How do I integrate expressions involving or resulting in hyperbolic functions?

• The following integration skills may be required
• Definite integration, area under a curve
• Reverse chain rule (‘adjust’ and ‘compensate’)
• Substitution
• Integration by parts
• Hyperbolic identities may be required to rewrite an expression into an integrable form
• For products involving ex and a hyperbolic function use the definition involving ex and e-x for the hyperbolic function to write everything in terms of exponentials

#### How do I integrate expressions involving inverse hyperbolic functions?

• To integrate inverse hyperbolic functions you would use integration by parts using the same technique as integrating lnx
• Write the functions as a product with 1
• e.g.
• Differentiate the inverse function and integrate 1 when integrating by parts

#### Exam Tip

• Be aware of what is given in the formula booklet
• Practise using it to find integrals
• The results for hyperbolic functions and the inverse circular trig functions are listed together so try not to get confused
• If you can't spot a relevant hyperbolic identity then using exponentials can make the expression easier and quicker to integrate

#### Worked Example

Find the following integrals:

a)

b)