Study guides › 9231 FP2 › Differentiation Maths4U Original · Learn, check, explain
Hyperbolic and inverse functions — 9231 FP2
Use chain rules and domains for inverse hyperbolic derivatives.
1. Understand 2. See examples 3. Check answers 4. Explain your working Notes and key methods Useful rules are \((\operatorname{arsinh}x)'=1/\sqrt{1+x^2}\), \((\operatorname{arcosh}x)'=1/\sqrt{x^2-1}\) for x>1, and \((\operatorname{artanh}x)'=1/(1-x^2)\) for |x|<1.
An inner function contributes its derivative. Inverse-function derivatives can also be derived from y=g⁻¹(x), x=g(y), then dy/dx=1/g′(y), with branch conditions retained.
Check the conditions Keep square-root and logarithm domains explicit. A reciprocal-function derivative is different from an inverse-function derivative. Three worked examples Example 1 · Hyperbolic and inverse-function derivatives Near x=0, let \(f(x)=5 \log{\left(x + 2 \right)} + \operatorname{asinh}{\left(3 x \right)}\). All derivatives below are evaluated at x=0.
Find f′(0). Find f″(0). Find f‴(0). Find f⁽⁴⁾(0). Show the reasoning (a) \(f'(x)=\frac{3}{\sqrt{9 x^{2} + 1}} + \frac{5}{x + 2}\); evaluate at zero.
Answer: \(\frac{11}{2}\).
(b) \(f''(x)=- (\frac{27 x}{\left(9 x^{2} + 1\right)^{\frac{3}{2}}} + \frac{5}{\left(x + 2\right)^{2}})\); evaluate at zero.
Answer: \(- \frac{5}{4}\).
(c) \(f^{(3)}(0)=- \frac{103}{4}\)
Answer: \(- \frac{103}{4}\).
(d) \(f^{(4)}(0)=- \frac{15}{8}\)
Answer: \(- \frac{15}{8}\).
Example 2 · Hyperbolic and inverse-function derivatives Near x=0, let \(f(x)=6 \log{\left(x + 2 \right)} + \operatorname{asinh}{\left(4 x \right)}\). All derivatives below are evaluated at x=0.
Find f′(0). Find f″(0). Find f‴(0). Find f⁽⁴⁾(0). Show the reasoning (a) \(f'(x)=\frac{4}{\sqrt{16 x^{2} + 1}} + \frac{6}{x + 2}\); evaluate at zero.
Answer: \(7\).
(b) \(f''(x)=- 2 \left(\frac{32 x}{\left(16 x^{2} + 1\right)^{\frac{3}{2}}} + \frac{3}{\left(x + 2\right)^{2}}\right)\); evaluate at zero.
Answer: \(- \frac{3}{2}\).
(c) \(f^{(3)}(0)=- \frac{125}{2}\)
Answer: \(- \frac{125}{2}\).
(d) \(f^{(4)}(0)=- \frac{9}{4}\)
Answer: \(- \frac{9}{4}\).
Example 3 · Hyperbolic and inverse-function derivatives Near x=0, let \(f(x)=7 \log{\left(x + 2 \right)} + \operatorname{asinh}{\left(5 x \right)}\). All derivatives below are evaluated at x=0.
Find f′(0). Find f″(0). Find f‴(0). Find f⁽⁴⁾(0). Show the reasoning (a) \(f'(x)=\frac{5}{\sqrt{25 x^{2} + 1}} + \frac{7}{x + 2}\); evaluate at zero.
Answer: \(\frac{17}{2}\).
(b) \(f''(x)=- (\frac{125 x}{\left(25 x^{2} + 1\right)^{\frac{3}{2}}} + \frac{7}{\left(x + 2\right)^{2}})\); evaluate at zero.
Answer: \(- \frac{7}{4}\).
(c) \(f^{(3)}(0)=- \frac{493}{4}\)
Answer: \(- \frac{493}{4}\).
(d) \(f^{(4)}(0)=- \frac{21}{8}\)
Answer: \(- \frac{21}{8}\).
Common mistakes Do not omit the inner derivative. Repeated derivatives must also differentiate the existing denominator or power. Practise with answer checking Original exam-style problems Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.
Next skill: Maclaurin series →
About these materials Prepared independently from learning objectives with AI assistance for Maths4U. Mathematical checks accompany the authored models; independent teacher review is pending. Maths4U marking guidance is not an official examination-board mark scheme. Existing questions keep their original sources and results.