DEFINITION
GEOMETRIC DEF: the slope (gradient) of tangent line
PHYSICAL DEF: instantaneous rate of change
!Geometric Defenition of Derivative Ex
看上图，Q无限接近于P时的$\frac{\Delta f}{\Delta x}$就是Tangent line的slope
所以我们可以得出下面这个公式：
[!NOTE]
$$
f'(x)=\lim_{ \Delta x \to 0 } \frac{f(x_{0}+\Delta x)-f(x)}{\Delta x}
$$
How to solve any Derivative?
使用Limit的方法
[!example] Example 1
$$
f(x)=\frac{1}{x}
$$
$$
\begin{align}
f'(x)=\frac{\Delta f}{\Delta x}=\frac{\left( \frac{1}{x_{0}+\Delta x}-\frac{1}{x_{0}} \right)}{\Delta x} \ \
直接把 \Delta x 趋近于零分子和分母都是0，所以先化简(通分)\ \
=\frac{1}{\Delta x}\left( \frac{x_{0}-(x_{0}+\Delta x)}{(x_{0}+\Delta x)x_{0}} \right)
=\frac{-1}{(x_{0}+\Delta x)x_{0}} \ \
这时候再把\Delta x无限趋近于0，得到 \ \
=\frac{-1}{x_{0}^2 }
\end{align}
$$
[!example] example 2
$$
f(x)=x^n
$$
$$
f'(x)=\frac{\Delta f}{\Delta x}=\frac{(x+\Delta x)^n-x^n}{\Delta x}
$$
用Binominal Theorem展开
$$
(x+\Delta x)^n=x^n+n(x^{x-1}\Delta x)+O((\Delta x)^2)
$$
$O((\Delta x)^2)$代表省略剩下的比$(\Delta x)^2$更高阶的项
$$
\begin{align}
=\frac{x^n+n(x^{x-1}\Delta x)+O((\Delta x)^2)-x^n}{\Delta x}\
=\frac{n(x^{x-1}\Delta x)+O((\Delta x)^2)}{\Delta x} \
=nx^{n-1}+O(\Delta x) \ \
把\Delta x无限趋近0 \
=nx^{n-1}
\end{align}
$$
由此，我们就推出了微分的最关键的公式
[!note] 关键公式
$$nx^{n-1}$$