求一个变量的对数对另一个变量对数的偏导

求一个变量的对数对另一个变量对数的偏导

今天看论文看到后面算法部分,发现了这样的问题,一时间懵住了,仔细一想才回过味来

已知:

\[F(x_i,x_j,\theta) = 0 \]

\[log x_j = g(log x_i,\theta) \]

求:

\[ \frac{\partial logx_i}{\partial logx_j} \]

解:

因为我们研究\(x_i\)和\(x_j\)的关系所以变量\(\theta\)的增量为0,\(d\theta = 0\)

所以

\[dF = \frac{\partial F}{\partial x_i}dx_i +\frac{\partial F}{\partial x_j}dx_j \]

\[dF = \frac{\partial F}{\partial x_i}\frac{d x_i}{dlogx_i}dlogx_i +\frac{\partial F}{\partial x_j}\frac{dx_j}{dlogx_j}dlogx_j \]

\[dF = \frac{\partial F}{\partial x_i}\frac{d x_i}{dlogx_i}dlogx_i +\frac{\partial F}{\partial x_j}\frac{dx_j}{dlogx_j}\frac{\partial log x_j}{\partial log x_i}dlogx_i \]

\[0 = \frac{\partial F}{\partial x_i}\frac{d x_i}{dlogx_i} +\frac{\partial F}{\partial x_j}\frac{dx_j}{dlogx_j}\frac{\partial log x_j}{\partial log x_i} \]

\[-\frac{\frac{\partial F}{\partial x_i}}{\frac{\partial F}{\partial x_j}} = \frac{\partial log x_j}{\partial log x_i}\frac{\frac{dx_j}{dlogx_j}}{\frac{d x_i}{dlogx_i}} \]

\[-\frac{\frac{\partial F}{\partial x_i}}{\frac{\partial F}{\partial x_j}} = \frac{\partial log x_j}{\partial log x_i}\frac{\frac{dlogx_i}{dx_i}}{\frac{d logx_j}{dx_j}} \]

\[-\frac{\frac{\partial F}{\partial x_i}}{\frac{\partial F}{\partial x_j}} = \frac{\partial log x_j}{\partial log x_i}\frac{x_j}{x_i} \]

\[-\frac{\frac{\partial F}{\partial x_i}}{\frac{\partial F}{\partial x_j}} \frac{x_i}{x_j}= \frac{\partial log x_j}{\partial log x_i} \]

\[\frac{\partial x_j}{\partial x_i} \frac{x_i}{x_j}= \frac{\partial log x_j}{\partial log x_i} \]