Hyperbolic power functions

Questions involving hyperbolic functions raised to powers (e.g., cosh^n x, sinh x sin x) where the derivative relationship involves recurrence relations or products of hyperbolic and trigonometric functions.

4 questions · Challenging +1.2

4.07d Differentiate/integrate: hyperbolic functions4.08a Maclaurin series: find series for function
Sort by: Default | Easiest first | Hardest first
Edexcel CP2 2022 June Q9
8 marks Challenging +1.2
9. $$y = \cosh ^ { n } x \quad n \geqslant 5$$
    1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = n ^ { 2 } \cosh ^ { n } x - n ( n - 1 ) \cosh ^ { n - 2 } x$$
    2. Determine an expression for \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
  2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form.
Edexcel CP2 Specimen Q5
10 marks Challenging +1.2
5. $$y = \sin x \sinh x$$
  1. Show that \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = - 4 y\)
  2. Hence find the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in its simplest form.
  3. Find an expression for the \(n\)th non-zero term of the Maclaurin series for \(y\).
SPS SPS FM Pure 2024 February Q15
8 marks Challenging +1.2
\(y = \cosh^n x\) \quad \(n \geq 5\)
    1. Show that $$\frac{d^2y}{dx^2} = n^2\cosh^n x - n(n-1)\cosh^{n-2}x$$ [4]
    2. Determine an expression for \(\frac{d^4y}{dx^4}\) [2]
  2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer. [2]
SPS SPS FM Pure 2026 November Q9
8 marks Challenging +1.3
Given that $$y = \cos x \sinh x \quad x \in \mathbb{R}$$
  1. show that $$\frac{d^4y}{dx^4} = ky$$ where \(k\) is a constant to be determined. [5]
  2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form. [3]