CAIE A-Level Further Mathematics 2.3.1 Hyperbolic Functions

Practise differentiating hyperbolic and inverse hyperbolic functions within explicit, implicit and parametric relationships.

Exam points

differentiate sinh, cosh and tanh using chain, product or quotient rules
derive inverse hyperbolic derivatives in radical or rational form
differentiate implicitly or parametrically, then substitute coordinates to verify a gradient

[Maximum number: 3]

Show that $\frac{\mathrm{d} y}{\mathrm{~d} x}=-\operatorname{cosech} x$ .

$\begin{aligned} \frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{\operatorname{sech}^{2}\left(\frac{1}{2} x\right)}{2 \tanh \left(\frac{1}{2} x\right)}=-\frac{1}{2 \sinh \left(\frac{1}{2} x\right) \cosh \left(\frac{1}{2} x\right)} \text { Or } \frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{\operatorname{cosech}^{2}\left(\frac{1}{2} x\right)}{2 \operatorname{coth}\left(\frac{1}{2} x\right)}=-\frac{1}{2 \sinh \left(\frac{1}{2} x\right) \cosh \left(\frac{1}{2} x\right)} \end{aligned}$

M1 A1

Uses chain rule.

$=-\frac{1}{\sinh (x)}=-\operatorname{cosech} x$

A1

AG

3

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2.3.1—Hyperbolic functions question 1