| Date | None Specimen | Marks available | 4 | Reference code | SPNone.1.sl.TZ0.7 |
| Level | SL only | Paper | 1 | Time zone | TZ0 |
| Command term | Find | Question number | 7 | Adapted from | N/A |
Question
Given that \(f(x) = \frac{1}{x}\) , answer the following.
Find the first four derivatives of \(f(x)\) .
[4]
a.
Write an expression for \({f^{(n)}}(x)\) in terms of x and n .
[3]
b.
Markscheme
\(f'(x) = - {x^{ - 2}}\) (or \( - \frac{1}{{{x^2}}}\) ) A1 N1
\(f''(x) = 2{x^{ - 3}}\) (or \(\frac{2}{{{x^3}}}\) ) A1 N1
\(f'''(x) = - 6{x^{ - 4}}\) (or \( - \frac{6}{{{x^4}}}\) ) A1 N1
\({f^{(4)}}(x) = 24{x^{ - 5}}\) (or \(\frac{{24}}{{{x^5}}}\) ) A1 N1
[4 marks]
a.
\({f^{(n)}}(x) = \frac{{{{( - 1)}^n}n!}}{{{x^{n + 1}}}}\) or \({( - 1)^n}n!({x^{ - (n + 1)}})\) A1A1A1 N3
[3 marks]
b.
Examiners report
[N/A]
a.
[N/A]
b.
Syllabus sections
Topic 6 - Calculus » 6.2 » Derivative of \({x^n}\left( {n \in \mathbb{Q}} \right)\) , \(\sin x\) , \(\cos x\) , \(\tan x\) , \({{\text{e}}^x}\) and \(\ln x\) .
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