Pearson Edexcel IAL Mathematics FP3.3.1 Differentiation of hyperbolic cosh2x

Practise differentiating hyperbolic functions such as tanh 3x and products involving sinh or cosh expressions.

[Maximum number: 5]

\mathrm{g}(x)=\arctan \left(\mathrm{e}^{2 x}\right)

Show that

\mathrm{g}^{\prime \prime}(x)=k \operatorname{sech}(2 x) \tanh (2 x)

where k is a constant to be found.

$g(x)=\arctan(e^{2x})$ $g'(x)=\frac{2e^{2x}}{e^{4x}+1}=\operatorname{sech}(2x)$ $g''(x)=-2\operatorname{sech}(2x)\tanh(2x)$
So k=-2.

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FP3.3.1 - Differentiation of hyperbolic cosh2x question 1