数学公式【合集】

泰勒展开式

$推导原理:麦克劳林公式f(x)=f(0)+f^{\prime }(0)x+......+\frac{f^{(n)}(0)x^n}{n!}+o(x^n)$

$助记：sinx,tanx分别与arcsinx,arctanx的x^3符号相反$

$1.sinx=x-\frac{1}{3!}{x}^3+\frac{1}{5!}{x}^5+o(x^5)$

$2.cosx=1-\frac{1}{2!}{x}^2+\frac{1}{4!}{x}^4+o(x^4)$

$3.tanx=x+\frac{1}{3!}{x}^3-\frac{1}{5!}{x}^5+o(x^5)$

$4.arcsinx=x+\frac{1}{6}{x}^3+o(x^3)$

$5.arctanx=x-\frac{1}{3}{x}^3+o(x^3)$

$6.ln(1+x)=x-\frac{1}{2}{x}^2+\frac{1}{3}{x}^3-\frac{1}{4}{x}^4+o(x^4)$

$7.e^x=1+x+\frac{1}{2}{x}^2+\frac{1}{3!}{x}^3+\cdot \cdot \cdot +\frac{1}{n!}{x}^n+o(x^n)$

$8.(1+x{)}^{\alpha }=1+\alpha x+\frac{\alpha (\alpha -1)}{2}{x}^2+o(x^2)$

$当x\to 0时：$
$$\begin{gathered} 1.变型： \\ \sqrt[\alpha ]{1+f(x)}-1=(1+f(x){)}^{\frac{1}{\alpha }}-1\sim \frac{1}{\alpha }f(x) \\ ln(f(x))=ln(1+(f(x)-1))\sim f(x)-1 \\ {e}^{f(x)}-e^{g(x)}=e^{g(x)}(e^{f(x)-g(x)}-1)=e^{g(x)}(f(x)-g(x)) \end{gathered}$$
$$\begin{gathered} 2.组合： \\ sinx-tanx=-\frac{1}{2}{x}^3+o(x^3) \\ arcsinx-arctanx=\frac{1}{2}{x}^3+o(x^3) \end{gathered}$$
$3.等价无穷小(等价无穷小是泰勒公式的一种特殊情况)$
$$\begin{gathered} sinx\sim x,ln(1+x)\sim x,tanx\sim x,e^x-1\sim x \\ arcsinx\sim x,arctanx\sim x,1-cosx\sim \frac{1}{x^2},(1+x{)}^{\alpha }-1\sim \alpha x \end{gathered}$$

求导公式

$助记:名称带有"余"字，如余弦cos、余割csc、余切cot，求导都带负号$

$(1)(C{)}^{\prime }=0$
$(2)(x^u{)}^{\prime }=u{x}^{u-1}$
$(3)(sinx{)}^{\prime }=cosx$
$(4)(cosx{)}^{\prime }=-sinx$
$(5)(tanx{)}^{\prime }=se{c}^{2}x=\frac{1}{co{s}^{2}x}$
$(6)(cotx{)}^{\prime }=-cs{c}^{2}x=-\frac{1}{si{n}^{2}x}$
$(7)(secx{)}^{\prime }=sectanx$
$(8)(cscx{)}^{\prime }=-csccotx$
$(9)(a^x{)}^{\prime }=a^{x}lna$
$(10)(e^x{)}^{\prime }=e^x$
$(11)(lo{g}_a^x{)}^{\prime }=\frac{1}{xlna}$
$(12)(lnx{)}^{\prime }=\frac{1}{x}$
$(13)(arcsinx{)}^{\prime }=\frac{1}{\sqrt{1-x^2}}$
$(14)(arccosx{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}}$
$(15)(arctanx{)}^{\prime }=\frac{1}{1+x^2}$
$(16)(arccotx{)}^{\prime }=-\frac{1}{1+x^2}$

$求导组合运算:$

$(1)(u+v{)}^{\prime }=u^{\prime }+v^{\prime }$
$(2)(Cu{)}^{\prime }=C{u}^{\prime }$
$(3)(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$
$(4)(\frac{u}{v}{)}^{\prime }=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$

$推导：$
$(tanx{)}^{\prime }=(\frac{sinx}{cosx}{)}^{\prime }=\frac{(sinx{)}^{\prime }cosx-sinx(cosx{)}^{\prime }}{co{s}^{2}x}=\frac{(co{s}^{2}x+si{n}^{2}x)}{co{s}^{2}x}=\frac{1}{co{s}^{2}x}=se{c}^{2}x$
$【简记分母co{s}^{2}x，cotx为tanx倒数则相反分母si{n}^{2}x】$

三角函数

奇变偶不变，符号看象限
$tan(\frac{\pi }{2}x+\frac{\pi }{2})=-cot\frac{\pi }{2}x$

$si{n}^2\alpha =\frac{1-cos2\alpha }{2}$
$co{s}^2\alpha =\frac{1+cos2\alpha }{2}$

$sin(\alpha +\beta )=sin\alpha cos\beta +cos\alpha sin\beta$
$cos(\alpha +\beta )=cos\alpha cos\beta +sin\alpha sin\beta$

$二倍角公式+万能公式（2\alpha \to \alpha ）:$
$sin2\alpha =2sin\alpha cos\alpha =\frac{2tan\alpha }{1+ta{n}^2\alpha }$
$cos2\alpha =co{s}^2\alpha -si{n}^2\alpha =\frac{1-ta{n}^2\alpha }{1+ta{n}^2\alpha }$
$tan2\alpha =\frac{2tan\alpha }{1-ta{n}^2\alpha }$

角α	0°	30°	45°	60°	90°
sinα	0	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	1
cosα	1	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	0
tanα	$0$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	$-$

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