Differentiating & Integrating Hyperbolic Functions (Edexcel A Level Further Maths: Core Pure)

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Paul

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5 February 2024

Differentiating Hyperbolic Functions

What are the derivatives of the hyperbolic functions?

$\frac{d}{d x} (\sinh x) = \cosh x$
$\frac{d}{d x} (\cosh x) = \sinh x$
$\frac{d}{d x} (\tanh x) = {sech}^{2} x$
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 e^x also being e^x giving rise to the repetition of terms

What are the derivatives of the inverse hyperbolic functions?

$\frac{d}{d x} (arsinh x) = \frac{1}{\sqrt{1 + x^{2}}}$
$\frac{d}{d x} (ar \cosh x) = \frac{1}{\sqrt{x^{2} - 1}}$ $, x > 1$
$\frac{d}{d x} (ar \tanh x) = \frac{1}{1 - x^{2}}$ $, | x | < 1$
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
- $y = arsinh x$ can be written $x = \sinh y$
STEP 2
Differentiate with respect to y
- $\frac{d x}{d y} = \cosh y$
STEP 3
Write the derivative in terms of x
- $\frac{d x}{d y} = \cosh y = \pm \sqrt{\sinh^{2} y + 1} = \pm \sqrt{x^{2} + 1}$
STEP 4
Take the reciprocal
- $\frac{d y}{d x} = \pm \frac{1}{\sqrt{x^{2} + 1}}$
STEP 5
Use the graph to determine whether it is positive of negative
- The graph of $y = arsinh x$ has a positive gradient everywhere
- $\frac{d y}{d x} = \frac{1}{\sqrt{x^{2} + 1}}$

Examiner 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

Given that

y = 2 x \sinh x

, show that

\frac{d^{2} y}{d x^{2}} = y + 4 \cosh x

RaTp7qLC_al-fm-4-1-4-we-solution-differentiation-a

Given that

f (x) = 3 arcosh 4 x

, show that

f' (5) = \frac{a}{\sqrt{b}}

where

a \in ℚ

and

b \in ℕ

are constants to be found.

al-fm-4-1-4-we-solution-differentiation-b

Integrating Hyperbolic Functions

What are the integrals of the hyperbolic functions?

These are the reverse results of the derivatives, remembering “+c” of course!
- $\int \sinh x d x = \cosh x + c$
- $\int \cosh x d x = \sinh x + c$
  - These are given in the integration section of the formulae booklet
- $\int {sech}^{2} x d x = \tanh x + c$
  - This can be deduced from the differentiation section of the formulae booklet
There is also the integral of tanhx
- $\int \tanh x d x = \ln \cosh x + c$
  - This is given in the integration section of the formulae booklet
  - It can be shown using the substitution $u = \cosh x$
- 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 e^x and a hyperbolic function use the definition involving e^x and e^-x for the hyperbolic function to write everything in terms of exponentials
- $\int e^{x} \cosh x d x = \frac{1}{2} \int e^{x} (e^{x} + e^{- x}) d x = \frac{1}{2} \int e^{2 x} + 1 d x$

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. $arsinh x = 1 \times arsinh x$
- Differentiate the inverse function and integrate 1 when integrating by parts
  - $\int arsinh x d x = x arsinh x - \int \frac{x}{\sqrt{x^{2} + 1}} d x = x arsinh x - \sqrt{x^{2} + 1} + c$

Examiner 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:

\int \sinh 3 x \cosh^{3} 3 x d x

XGrWSm1c_al-fm-4-1-4-we-solution-integration-a

\int \cosh^{4} x - \sinh^{4} x d x

al-fm-4-1-4-we-solution-integration-b

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