📝 题目
例 23 设 $y = {x}^{a}$ ,求 ${y}^{\left( n\right) }$ .
💡 答案与解析
解
$$ {y}^{\prime } = \alpha {x}^{\alpha - 1}, $$
$$ {y}^{\prime \prime } = \alpha \left( {\alpha - 1}\right) {x}^{\alpha - 2}, $$
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$$ {y}^{\left( n\right) } = \alpha \left( {\alpha - 1}\right) \cdots \left( {\alpha - n + 1}\right) {x}^{\alpha - n}. $$
如果 $\alpha = m \in \mathbb{N}$ ,那么 $n > m$ 时就有 ${y}^{\left( n\right) } = 0$ .