Discussion
If \(f(x) = \dfrac{1}{x^{2}}\), then the first divided difference [a, b] is :
Answer: A
The divided difference is −(a + b)/(a2b2).
\[ \begin{aligned} f[a, b] &= \frac{f(b) - f(a)}{b - a} = \frac{\frac{1}{b^{2}} - \frac{1}{a^{2}}}{b - a} \\ &= \frac{a^{2} - b^{2}}{a^{2}b^{2}(b - a)} \\ &= \frac{-(a + b)}{a^{2}b^{2}} \end{aligned} \]
since \(a^{2} - b^{2} = -(b - a)(a + b)\). The option with \(+(a + b)\) has the sign wrong. The options \((a + b)/ab\) and \((a - b)/ab\) lack the squares in the denominator that \(1/x^{2}\) produces.
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