Latex：导数【高中常用公式】

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大学、高中、初中、小学常用公式，一键模板。

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导数的定义：

{f \prime { \left( {\mathop{{x}}\nolimits_{{0}}} \right) }=\mathop{{ \text{lim} }}\limits_{{ \Delta x \to 0}}\frac{{ \Delta y}}{{ \Delta x}}=\mathop{{ \text{lim} }}\limits_{{ \Delta x \to 0}}\frac{{f{ \left( {\mathop{{x}}\nolimits_{{0}}+ \Delta x} \right) }-f{ \left( {\mathop{{x}}\nolimits_{{0}}} \right) }}}{{ \Delta x}}}

导数的记法：

\mathop{{\left. y \prime \right| }}\nolimits_{{x=\mathop{{x}}\nolimits_{{0}}}}

莱布尼兹记法：

\begin{array}{*{20}{l}}

{\mathop{{\left. \frac{{ \text{d} y}}{{ \text{d} x}} \right| }}\nolimits_{{x=\mathop{{x}}\nolimits_{{0}}}}}\\

{\mathop{{\left. \frac{{ \text{d} f{ \left( {x} \right) }}}{{ \text{d} x}} \right| }}\nolimits_{{x=\mathop{{x}}\nolimits_{{0}}}}}

\end{array}

牛顿记法：

\mathop{{\left. \dot {y} \right| }}\nolimits_{{x=\mathop{{x}}\nolimits_{{0}}}}

反函数求导法则：

\left[ {\mathop{{f}}\nolimits^{{-1}}{ \left( {x} \right) }} \left] \prime =\frac{{1}}{{f \prime { \left( {y} \right) }}}\right. \right.

复合函数求导法则：

\begin{array}{*{20}{l}}

{y=f{ \left( {u} \right) },u=g{ \left( {x} \right) }}\\

{\frac{{ \text{d} y}}{{ \text{d} x}}=\frac{{ \text{d} y}}{{ \text{d} u}} \cdot \frac{{ \text{d} u}}{{ \text{d} x}}}

\end{array}

和差积商求导法则：

\begin{array}{*{20}{l}}

{ \left( {u \pm v} \left) \prime ={u \prime } \pm {v \prime }\right. \right. }\\

{ \left( {Cu} \left) \prime =C{u \prime }\right. \right. }\\

{ \left( {uv} \left) \prime ={u \prime }v+u{v \prime }\right. \right. }\\

{ \left( {\frac{{u}}{{v}}} \left) \prime =\frac{{u \prime v-u{v \prime }}}{{\mathop{{v}}\nolimits^{{2}}}},{ \left( {v \neq 0} \right) }\right. \right. }

\end{array}

基本导数1：

\begin{array}{*{20}{l}}

{ \left( {C} \left) \prime =0\right. \right. }\\

{ \left( {\mathop{{x}}\nolimits^{{ \mu }}} \left) \prime = \mu \mathop{{x}}\nolimits^{{ \mu -1}}\right. \right. }

\end{array}

基本导数2：

\begin{array}{*{20}{l}}

{ \left( { \text{sin} x} \left) \prime = \text{cos} x\right. \right. }\\

{ \left( { \text{cos} x} \left) \prime =- \text{sin} x\right. \right. }\\

{ \left( { \text{tan} x} \left) \prime =\mathop{{ \text{sec} }}\nolimits^{{2}}x\right. \right. }\\

{ \left( { \text{cot} x} \left) \prime =-\mathop{{ \text{csc} }}\nolimits^{{2}}x\right. \right. }\\

{ \left( { \text{sec} x} \left) \prime = \text{sec} x \text{tan} x\right. \right. }\\

{ \left( { \text{csc} x} \left) \prime =- \text{csc} x{ \text{cot} x}\right. \right. }

\end{array}

基本导数3：

\begin{array}{*{20}{l}}

{ \left( {\mathop{{a}}\nolimits^{{x}}} \left) \prime =\mathop{{a}}\nolimits^{{x}} \text{ln} a\right. \right. }\\

{ \left( {\mathop{{e}}\nolimits^{{x}}} \left) \prime =\mathop{{e}}\nolimits^{{x}}\right. \right. }\\

{ \left( {\mathop{{ \text{log} }}\nolimits_{{a}}x} \left) \prime =\frac{{1}}{{x \text{ln} a}}\right. \right. }\\

{ \left( { \text{ln} a} \left) \prime =\frac{{1}}{{x}}\right. \right. }

\end{array}

基本导数4：

\begin{array}{*{20}{l}}

{ \left( { \text{arcsin} x} \left) \prime =\frac{{1}}{{\sqrt{{1-\mathop{{x}}\nolimits^{{2}}}}}}\right. \right. }\\

{ \left( { \text{arccos} x} \left) \prime =-\frac{{1}}{{\sqrt{{1-\mathop{{x}}\nolimits^{{2}}}}}}\right. \right. }\\

{ \left( { \text{arctan} x} \left) \prime =\frac{{1}}{{1+\mathop{{x}}\nolimits^{{2}}}}\right. \right. }\\

{ \left( { \text{arccot} x} \left) \prime =-\frac{{1}}{{1+\mathop{{x}}\nolimits^{{2}}}}\right. \right. }

\end{array}