📝 题目
例 25 设 $y = \ln \left( {1 + x}\right)$ ,求 ${y}^{\left( n\right) }$ .
💡 答案与解析
解 ${y}^{\prime } = {\left( 1 + x\right) }^{-1} = \frac{1}{1 + x}$ .
$$ {y}^{\left( n\right) } = {\left( {\left( 1 + x\right) }^{-1}\right) }^{\left( n - 1\right) } $$
$$ = \left( {-1}\right) \left( {-2}\right) \cdots \left( {-\left( {n - 1}\right) }\right) {\left( 1 + x\right) }^{-n} $$
$$ = {\left( -1\right) }^{n - 1}\frac{\left( {n - 1}\right) !}{{\left( 1 + x\right) }^{n}}. $$