# 判断二元函数是否可微的定义公式太长记不住？其实你已经记住了！

## 一、前言

$$\textcolor{orange}{ \lim \limits_{\substack{\Delta x \rightarrow 0 \\ \Delta y \rightarrow 0}} \frac{[f(x_{0} + \Delta x, y_{0} + \Delta y) – f(x_{0}, y_{0})] – [f^{\prime}_{x}(x_{0}, y_{0}) \Delta x + f^{\prime}_{y}(x_{0}, y_{0}) \Delta y]}{\sqrt{(\Delta x)^{2} + (\Delta y)^{2}}} }$$

## 二、正文

$$\lim_{\Delta x \rightarrow 0} \frac{f(x_{0} + \Delta x) – f(x_{0})}{\Delta x} = f^{\prime}(x_{0})$$

$$\lim_{\Delta x \rightarrow 0} \frac{f(x_{0} + \Delta x) – f(x_{0})}{\Delta x} = f^{\prime}(x_{0}) \Rightarrow$$

$$\lim_{\Delta x \rightarrow 0} \frac{f(x_{0} + \Delta x) – f(x_{0})}{\Delta x} = \frac{f^{\prime}(x_{0}) \Delta x}{\Delta x} \Rightarrow$$

$$\lim_{\Delta x \rightarrow 0} \frac{f(x_{0} + \Delta x) – f(x_{0})}{\Delta x} – \frac{f^{\prime}(x_{0}) \Delta x}{\Delta x} = 0 \Rightarrow$$

$$\lim_{\Delta x \rightarrow 0} \frac{[f(x_{0} + \Delta x) – f(x_{0}) ] – [ f^{\prime}(x_{0}) \Delta x] }{\Delta x} = 0 \Rightarrow$$

$$\lim_{\Delta x \rightarrow 0} \frac{[f(x_{0} + \Delta x) – f(x_{0}) ] – [ f^{\prime}(x_{0}) \Delta x] }{\sqrt{(\Delta x)^{2}}} = 0 \tag{1}$$

$$\lim \limits_{\substack{\Delta x \rightarrow 0 \\ \Delta y \rightarrow 0}} \frac{[f(x_{0} + \Delta x, \textcolor{red}{y_{0} + \Delta y}) – f(x_{0}, \textcolor{red}{y_{0}})] – [f^{\prime}_{x}(x_{0}, \textcolor{red}{y_{0}}) \Delta x + \textcolor{red}{f^{\prime}_{y}(x_{0}, y_{0}) \Delta y}]}{\sqrt{(\Delta x)^{2} + \textcolor{red}{(\Delta y)^{2}}}}$$

Next

$$\Delta z = A \Delta x+B \Delta y+o(\rho)$$

$$\textcolor{orange}{ f\left(x_{0}+\Delta x, y_{0}+\Delta y\right)-f\left(x_{0}, y_{0}\right)= A \Delta x+B \Delta y+o(\rho)} \tag{1}$$

$$\frac{A \Delta x+B \Delta y+o(\rho)-[A \Delta x+B \Delta y]}{\rho} = \frac{o(\rho)}{\rho}=0$$