Derivatives - 冰芹的 Notes

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：

$$ f'(x)=\lim_{ \Delta x \to 0 } \frac{f(x_{0}+\Delta x)-f(x)}{\Delta x} $$

How to solve any Derivative?

使用Limit的方法

Example 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 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

关键公式

$$nx^{n-1}$$