碎片记录

导数定义

• $f'(x_{0}) = \lim \limits_{\triangle x \to 0} \dfrac{f(x_{0} + \triangle x) - f(x_{0})}{\triangle x}$

• $f'(x_{0}) = \lim \limits_{h \to 0} \dfrac{f(x_{0} + h) - f(x_{0})}{h}$

• $f'(x_{0}) = \lim \limits_{x \to x_{0}} \dfrac{f(x) - f(x_{0})}{x - x_{0}}$

微分定义

• $\triangle y = A \triangle x + o(\triangle x) = dy + o(\triangle x) \\ (\triangle x \rightarrow 0，A 为常数，A \triangle x 称为 \triangle y 的线性主部，o(\triangle x) 是 \triangle x 的高阶无穷小）$

• $dy = A \triangle x = f'(x) dx，(dx 等价于 \triangle x)$

常用极限

• $\lim \limits_{x \to 0} \dfrac{sinx}{x} = 1$

• $\lim \limits_{x \to \infty}(1 + \dfrac{1}{x})^{x} = e$

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• $\lim \limits_{x \to 0}(1 + x)^{\frac{1}{x}} = e$

• $\lim \limits_{x \to \infty}(1 - \dfrac{1}{x})^{x} = \dfrac{1}{e}$

• $\lim \limits_{x \to \infty}(1 + \dfrac{a}{x})^{bx} = e^{ab}$

• $\lim \limits_{x \to \infty}(1 + \dfrac{a}{x})^{bx + c} = e^{ab}$

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• $\lim \limits_{n \to \infty} \sqrt[n]{n} = 1$

• $\lim \limits_{x \to 0} \dfrac{a^x - 1}{x} = lna$

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• $\lim \limits_{x \to \infty} \dfrac{a_{0}x^{n} + a_{1}x^{n - 1} + \cdots + a_{n}}{b_{0}x^{m} + b_{1}x^{m - 1} + \cdots + b_{m}} = \begin{cases} 0， \space n < m， \\ \dfrac{a_{0}}{b_0}，\space n = m \space (a_{0}b_{0} ≠ 0) \\ \infty，\space n > m \end{cases}$

常用导数

• $(\sqrt{x})' = \dfrac{1}{2\sqrt{x}}$

• $(e^{x})' = e^{x}$

• $(lnx)' = \dfrac{1}{x}$

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• $(x^{a})' = ax^{a - 1}$

• $(a^{x})' = a^{x}lna$

• $(log_{a}x)' = \dfrac{1}{xlna}$

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• $(ln|x|)' = \dfrac{1}{x}$

• $(x^{x})' = x^{x}(1 + lnx)$

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• $(sinx)' = cosx$
• $(cosx)' = -sinx$
• $(tanx)' = sec^{2}x$

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• $(cotx)' = -csc^{2}x$
• $(secx)' = secxtanx$
• $(cscx)' = -cscxcotx$

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• $(arcsinx)' = \dfrac{1}{\sqrt{1 - x^{2}}}$

• $(arccosx)' = -\dfrac{1}{\sqrt{1 - x^{2}}}$

• $(arctanx)' = \dfrac{1}{1 + x^{2}}$

• $(arccotx)' = -\dfrac{1}{1 + x^{2}}$

常用不定积分

• $\int kdx = kx + C$（k 为常数）

• $\int e^{x}dx = e^{x} + C$

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• $\int x^{a}dx = \dfrac{1}{a + 1}x^{a + 1} + C$（a ≠ -1）

• $\int a^{x}dx = \dfrac{a^{x}}{lna} + C$（a > 0，a ≠ 1）

• $\int \dfrac{1}{x}dx = ln|x| + C$（x ≠ 0）

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• $\int sinxdx = -cosx + C$

• $\int cosxdx = sinx + C$

• $\int tanxdx = -ln|cosx| + C$

• $\int cotxdx = ln|sinx| + C$

• $\int secxdx = ln|secx + tanx| + C$

• $\int cscxdx = ln|cscx - cotx| + C$

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• $\int sec^{2}xdx = tanx + C$

• $\int csc^{2}xdx = -cotx + C$

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• $\int secxtanxdx = secx + C$

• $\int cscxcotxdx = -cscx + C$

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• $\int \dfrac{1}{1 + x^{2}}dx = arctanx + C$

• $\int \dfrac{1}{a^{2} + x^{2}}dx = \dfrac{1}{a}arctan\dfrac{x}{a} + C$（a ≠ 0）

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• $\int \dfrac{1}{\sqrt{1 - x^{2}}}dx = arcsinx + C$

• $\int \dfrac{1}{\sqrt{a^{2} - x^{2}}}dx = arcsin\dfrac{x}{a} + C$（a > 0）

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• $\int \dfrac{1}{a^{2} - x^{2}}dx = \dfrac{1}{2a}ln\left| \dfrac{a + x}{a - x} \right| + C$

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• $\int \dfrac{1}{\sqrt{x^{2} \pm a^{2}}}dx = ln|x + \sqrt{x^{2} \pm a^{2}}| + C$

常用等价无穷小

当 x → 0 时，

• $x \sim sinx \sim tanx \sim arcsinx \sim arctanx \sim ln(1 + x) \sim e^{x} - 1$ 重点

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• $1 - cosx \sim \dfrac{1}{2} x^{2}$ 重点

• $1 - cos^{α}x \sim \dfrac{α}{2} x^{2}$

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• $a^{x} - 1 \sim xlna$

• $(1 + x)^{a} - 1 \sim ax$ 重点

• $log_{a}(1 + x) \sim \dfrac{x}{lna}$

• $ln(1 + x) - x \sim -\dfrac{1}{2} x^{2}$

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• $x - sinx \sim \dfrac{1}{6} x^{3}$

• $x - tanx \sim -\dfrac{1}{3} x^{3}$

• $x - arcsinx \sim -\dfrac{1}{6} x^{3}$

• $x - arctanx \sim \dfrac{1}{3} x^{3}$

• $tanx - sinx \sim \dfrac{1}{2} x^{3}$