反函数求导

四个基本公式：

\begin{array}{r} y_{x}^{'} = \frac{d y}{d x} = f^{'} (x), y_{x x}^{''} = \frac{d^{2} y}{d x^{2}} = f^{''} (x) \\ x_{y}^{'} = \frac{d x}{d y} = g^{'} (y), x_{y y}^{''} = \frac{d^{2} x}{d y^{2}} = g^{''} (y) \end{array}

我们可以通过如下过程得到二阶反函数的导数：

\begin{aligned} y_{x}^{'} = \frac{d y}{d x} = \frac{1}{\frac{d x}{d y}} = \frac{1}{x_{y}^{'}} \\ y_{x x}^{''} = \frac{d y_{x}^{'}}{d x} = \frac{d y_{x}^{'}}{d y} \cdot \frac{d y}{d x} = \frac{d (\frac{1}{x_{y}^{'}})}{d y} \cdot \frac{1}{x_{y}^{'}} = - \frac{x_{y y}^{''}}{x_{y}^{' 3}} \\ ⟹ x_{y y}^{''} = - \frac{y_{x x}^{''}}{x_{y}^{' 3}} \end{aligned}