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

Differentiating hyperbolic functions

What are the derivatives of the hyperbolic functions?

• $\frac{d}{dx}\left(\sinh x\right)=\cosh x$

• $\frac{d}{dx}\left(\cosh x\right)=\sinh x$

• $\frac{d}{dx}\left(\tanh x\right)={\mathrm{sech}}^{2}x$

• 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?

• $\frac{d}{dx}\left(\mathrm{arsinh}x\right)=\frac{1}{\sqrt{1+x^2}}$

• $\frac{d}{dx}\left(\mathrm{ar}\cosh x\right)=\frac{1}{\sqrt{x^2-1}}$$,x>1$

• $\frac{d}{dx}\left(\mathrm{ar}\tanh x\right)=\frac{1}{1-x^2}$, |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

  • $y=\mathrm{arsinh}x$ can be written $x=\sinh y$

• STEP 2

  Differentiate with respect to y

  • $\frac{dx}{dy}=\cosh y$

• STEP 3

  Write the derivative in terms of x

  • $\frac{dx}{dy}=\cosh y=\pm \sqrt{{\sinh }^{2}y+1}=\pm \sqrt{x^2+1}$

• STEP 4

  Take the reciprocal

  • $\frac{dy}{dx}=\pm \frac{1}{\sqrt{x^2+1}}$

• STEP 5

  Use the graph to determine whether it is positive of negative

  • The graph of $y=\mathrm{arsinh}x$ has a positive gradient everywhere

  • $\frac{dy}{dx}=\frac{1}{\sqrt{x^2+1}}$

Integrating hyperbolic functions

What are the integrals of the hyperbolic functions?

• These are the reverse results of the derivatives, remembering “+c” of course!

  • $\int \sinh xdx=\cosh x+c$

  • $\int \cosh xdx=\sinh x+c$ 

    • These are given in the integration section of the formulae booklet

  • $\int {\mathrm{sech}}^{2}xdx=\tanh x+c$

    • This can be deduced from the differentiation section of the formulae booklet

• There is also the integral of tanhx

  • $\int \tanh xdx=\ln \cosh x+c$ 

    • 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

  • $\int {\mathrm{e}}^x\cosh xdx=\frac{1}{2}\int {\mathrm{e}}^x\left({\mathrm{e}}^x+{\mathrm{e}}^{-x}\right)dx=\frac{1}{2}\int {\mathrm{e}}^{2x}+1dx$

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. $\mathrm{arsinh}x=1\times \mathrm{arsinh}x$

  • Differentiate the inverse function and integrate 1 when integrating by parts

    • $\int \mathrm{arsinh}xdx=x\mathrm{arsinh}x-\int \frac{x}{\sqrt{x^2+1}}dx=x\mathrm{arsinh}x-\sqrt{x^2+1}+c$