\frac{1}{\cosh^2x}=\operatorname{sech}^2x

Why does 2\sinh x-3\cosh x=0 have no solution?

It gives \tanh x=\frac32, but -1<\tanh x<1.

Calculus of hyperbolic functionsEdexcel 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)
$\frac{d}{dx}\tanh x$ ?: $\frac{1}{\cosh^2x}=\operatorname{sech}^2x$
$\frac{d}{dx}\tanh3x$ ?: $\frac{3}{\cosh^23x}$
$\frac{d}{dx}\cosh3x$ ?: $3\sinh3x$
$\frac{d}{dx}\left(x\sinh^2x\right)$ ?: $\sinh^2x+2x\sinh x\cosh x$
$\int\sinh x\,dx$ ?: $\cosh x+c$
$\int\cosh x\,dx$ ?: $\sinh x+c$
$\int\cosh kx\,dx$ ?: $\frac1k\sinh kx+c$
$\int\frac{1}{\cosh^2x}\,dx$ ?: $\tanh x+c$
$\int\frac{\cosh2x}{\sqrt{1+\sinh2x}}\,dx$ ?: $\sqrt{1+\sinh2x}+c$
Exact values of $\sinh(\ln2)$ and $\cosh(\ln2)$ ?: $\frac34$ and $\frac54$
Why does $2\sinh x-3\cosh x=0$ have no solution?: It gives $\tanh x=\frac32$ , but $-1<\tanh x<1$ .

Exam questions on Calculus of hyperbolic functions

Let $f(x)=\cosh3x$ .
Find the exact gradient of the curve $y=f(x)$ at the point where $x=\frac13\ln2$ .2 marks
The curve $C$ has equation $y=\tanh3x$ .
Find the equation of the tangent to $C$ at the origin.2 marks
The curve $C$ has equation $y=x\sinh^2x$ .
Find $\frac{dy}{dx}$ .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).