Discussion
The third divided difference of the function \(f(x) = 1/x\) with arguments \(p, q, r, s\) is :
Answer: C
The third divided difference is \(-\dfrac{1}{pqrs}\).
For \(f(x) = 1/x\):
\[ \begin{aligned} f[p, q] &= \frac{1/q - 1/p}{q - p} = -\frac{1}{pq} \\ f[p, q, r] &= \frac{-\frac{1}{qr} + \frac{1}{pq}}{r - p} = \frac{1}{pqr} \\ f[p, q, r, s] &= -\frac{1}{pqrs} \end{aligned} \]
The pattern is \(f[x_{0}, \ldots, x_{n}] = (-1)^{n}/(x_{0}\cdots x_{n})\). \(1/pqr\) is the second divided difference, \(1/pq\) the first with its sign dropped, and \(-s/pqr\) does not follow the pattern.
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