\operatorname{sech}x in terms of \cosh x?

\operatorname{sech}x=\frac{1}{\cosh x}

\operatorname{sech}^2x in terms of \tanh x?

\operatorname{sech}^2x=1-\tanh^2x

Solve \sinh x=2 without a calculator function.

Put u=e^x: u^2-4u-1=0, so x=\ln(2+\sqrt5).

Differentiating hyperbolic functionsEdexcel International A Level Further Maths: Flashcards

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$\frac{d}{dx}\sinh x$?

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$\frac{d}{dx}\sinh x$ ?: $\cosh x$
$\frac{d}{dx}\cosh x$ ?: $\sinh x$ (no minus sign)
$\frac{d}{dx}\tanh x$ ?: $\operatorname{sech}^2x$
$\operatorname{sech}x$ in terms of $\cosh x$ ?: $\operatorname{sech}x=\frac{1}{\cosh x}$
$\operatorname{sech}^2x$ in terms of $\tanh x$ ?: $\operatorname{sech}^2x=1-\tanh^2x$
$\frac{d}{dx}\sinh kx$ ?: $k\cosh kx$
$\frac{d}{dx}\cosh kx$ ?: $k\sinh kx$
$\frac{d}{dx}\tanh3x$ ?: $3\operatorname{sech}^23x$
$\frac{d}{dx}\sinh^2x$ ?: $2\sinh x\cosh x=\sinh2x$
$\frac{d}{dx}\left(x\sinh^2x\right)$ ?: $\sinh^2x+2x\sinh x\cosh x$
Which identity proves the derivative of $\tanh x$ ?: $\cosh^2x-\sinh^2x=1$
Solve $\sinh x=2$ without a calculator function.: Put $u=e^x$ : $u^2-4u-1=0$ , so $x=\ln(2+\sqrt5)$ .
Why reject a negative root when solving for $e^x$ ?: $e^x>0$ for all real $x$ .

Exam questions on Differentiating hyperbolic functions

The hyperbolic functions are defined by $\sinh x=\frac{e^x-e^{-x}}{2}$ , $\cosh x=\frac{e^x+e^{-x}}{2}$ and $\tanh x=\frac{\sinh x}{\cosh x}$ , and $\operatorname{sech}x=\frac{1}{\cosh x}$ .
Use the quotient rule to show that $\frac{d}{dx}\left(\tanh x\right)=\operatorname{sech}^2x$ .2 marks
The curve $C$ has equation $y=x\sinh^2x$ .
Show that $C$ has exactly one stationary point.2 marks
The function $f$ is defined by $f(x)=\frac{\cosh2x}{x+1}$ for $x>-1$ .
Find $f'(x)$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).