Derivative

泰勒公式

\[f(x) = \frac{f(x_0)}{0!} + \frac{f^{\prime}(x_0)}{1!}(x- x_0) + \frac{f^{\prime{\prime}}(x_0)}{2!}(x- x_0)^2 + \cdots + \frac{f^{(n)}(x_0)}{n!}(x- x_0)^n + R_n(x)\]

so that

\[\begin{split}\begin{align} e^{x} &= 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \cdots \\ \sin{x} &= x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 + \cdots \\ \cos{x} & = 1 - \frac{1}{2!}x^2 + \frac{1}{4!}x^4+ \cdots \\ \end{align}\end{split}\]

求导基础公式

求导运算法则

复合函数求导法则-链式法则