对数求导法

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适用条件

1.幂指函数, 例如: $ \ y=x^{\sin{x}}$

2.多因子乘幂型函数, 例如：
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\(y = \sqrt{x^{2}(1-x^{2})\sin x}\)
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\(y = a^{5}b^{6}c^{7}\)
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方法：

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step1：方程两边同时取对数
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step2：两边同时对\(x\)求导
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step3:计算关于\(y'\)的方程
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具体过程:

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已知\(y = f(x), 求y'\)
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两边同取对数: \(\ln{y} = \ln{f(x)}\)
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则: \((\ln{y})' = \ln{f'(x)}\)
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设\(\ln{y} = u\), 则 \(u(y) = \ln{y},y = (y)\)
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由链式法则得出：
\((\ln{y})' = [u(y)]' = u' \cdot y'\)
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根据常见函数求导公式: \(u' = (\log_{e}{y})' = \frac{1}{y}\)
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\(\therefore [u(y)]' = (\ln{y})' = \frac{1}{y} \cdot y'\)
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\(\therefore \frac{1}{y} \cdot y' = \ln{f'(x)}\)
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\(y' = \ln{f'(x)} \cdot y\)