# 微分

## 斜率同導數

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

${\displaystyle {\frac {\Delta f(x)}{\Delta x}}={\frac {f(x+\Delta x)-f(x)}{(x+\Delta x)-x}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$

${\displaystyle {dy \over dx}=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$

{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {(x+\Delta x)^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {x^{2}+2x\Delta x+(\Delta x)^{2}-x^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}{\frac {2x\Delta x+(\Delta x)^{2}}{\Delta x}}\\&=\lim _{\Delta x\to 0}2x+\Delta x\\&=2x\end{aligned}}}

## 偏微分

${\displaystyle y}$ 係常數，得到 ${\displaystyle {\frac {\partial z}{\partial x}}=2x+y}$

${\displaystyle x}$ 係常數，得到 ${\displaystyle {\frac {\partial z}{\partial y}}=2y+x}$

## 微分式

• 如果 ${\displaystyle y=x+c}$，咁 ${\displaystyle y}$ 嘅微分${\displaystyle {dy \over dx}=1}$
• 若果${\displaystyle y=ax^{n}}$${\displaystyle {dy \over dx}=anx^{n-1}}$
• ${\displaystyle {d \over dx}(au+bv)={d \over dx}(au)+{d \over dx}(bv)=a{du \over dx}+b{dv \over dx}}$
• ${\displaystyle {d \over dx}(uv)=u\cdot {dv \over dx}+v\cdot {du \over dx}}$
• ${\displaystyle {d \over dx}\left({\frac {u}{v}}\right)={\frac {v\cdot {du \over dx}-u\cdot {dv \over dx}}{v^{2}}}}$
• 鏈式法則：${\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}}$
• ${\displaystyle {d \over dx}(e^{x})=e^{x}}$
• ${\displaystyle {d \over dx}\ln(x)={1 \over x}}$