微分常用公式
$$(f(x) + g(x))' = f'(x) + g'(x)$$
$$(f(x) - g(x))' = f'(x) - g'(x)$$
$$(cf(x))' = c f'(x) \quad \text{（c為常數）}$$
$$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \quad \text{（乘法法則）}$$
$$\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \quad \text{（除法法則）}$$
$$(f(g(x)))' = f'(g(x)) \cdot g'(x) \quad \text{（鏈式法則）}$$
$$\frac{d}{dx}(x^n) = nx^{n-1}$$
$$\frac{d}{dx}(\sin x) = \cos x$$
$$\frac{d}{dx}(\cos x) = -\sin x$$
$$\frac{d}{dx}(\tan x) = \sec^2 x$$
$$\frac{d}{dx}(e^x) = e^x$$
$$\frac{d}{dx}(\ln x) = \frac{1}{x}$$
積分常用公式
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$
$$\int \frac{1}{x} dx = \ln|x| + C$$
$$\int e^x dx = e^x + C$$
$$\int \sin x , dx = -\cos x + C$$
$$\int \cos x , dx = \sin x + C$$
$$\int \sec^2 x , dx = \tan x + C$$
泰勒展開式
【给我俩分钟，还你泰勒公式记忆一片天空】 https://www.bilibili.com/video/BV1bC4y1H7cp/?share_source=copy_web&vd_source=b38b089ffe2c29ef61bebcff0f458ed8
其實只需要記住
$e^x$
$sin x$
$ln(1+x)$
這三個就可以
$$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots + \frac{x^n}{n!} + o(x^n) $$
$$ \sin x = x - \frac{x^3}{3!} + \cdots + \frac{(-1)^n}{(2n+1)!} x^{2n+1} + o(x^{2n+1}) $$
$$ \cos x = 1 - \frac{x^2}{2!} + \cdots + \frac{(-1)^n}{(2n)!} x^{2n} + o(x^{2n}) $$
$$ \frac{1}{1 - x} = 1 + x + x^2 + x^3 + \cdots + x^n + o(x^n) $$
$$ \frac{1}{1 + x} = 1 - x + x^2 - x^3 + \cdots + (-1)^n x^n + o(x^n) $$
$$ \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + \frac{(-1)^{n-1}}{n} x^n + o(x^n) $$