 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability-resit/6/a/solution

The displayed rational expression has domain

R ∖ {0}

; its value at zero is undefined despite the printed

R \to R

notation. On that domain, the power rule gives

f^{'} (x) f^{''} (x) f^{(3)} (x) f^{(4)} (x) f^{(5)} (x) = 2 x - 4 + \frac{4}{x ^{2}} - \frac{2}{x ^{3}}, = 2 - \frac{8}{x ^{3}} + \frac{6}{x ^{4}}, = \frac{24}{x ^{4}} - \frac{24}{x ^{5}}, = - \frac{96}{x ^{5}} + \frac{120}{x ^{6}}, = \frac{480}{x ^{6}} - \frac{720}{x ^{7}} .

These derivatives are smooth on every interval avoiding zero, including all intervals used below.

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