Hint
Keep square-root and logarithm domains explicit. Differentiate arsinh(ax) with the inner factor a.
Worked solution
(a)
\(\displaystyle f'(x)=\frac{8}{\sqrt{64 x^{2} + 1}} + \frac{10}{x + 2}\); evaluate at zero.
Answer: \(13\).
Answer: \(13\)
(b)
\(\displaystyle f''(x)=- 2 \left(\frac{256 x}{\left(64 x^{2} + 1\right)^{\frac{3}{2}}} + \frac{5}{\left(x + 2\right)^{2}}\right)\); evaluate at zero.
Answer: \(\displaystyle - \frac{5}{2}\).
Answer: \(\displaystyle - \frac{5}{2}\)
(c)
\(\displaystyle f^{(3)}(0)=- \frac{1019}{2}\)
Answer: \(\displaystyle - \frac{1019}{2}\).
Answer: \(\displaystyle - \frac{1019}{2}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Differentiate arsinh(ax) with the inner factor a.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not omit the inner derivative.
(b) · 2 marks
- 1 mark: Differentiate both terms a second time.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Repeated derivatives must also differentiate the existing denominator or power.
(c) · 2 marks
- 1 mark: Differentiate the power and logarithmic terms once more.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not omit the inner derivative.
Total: 6 marks.