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# 微分　x^α

$3\displaystyle{{ \( {x}^{\alpha } \) }^{\prime }=\alpha {x}^{ \alpha - 1 }}$

## ■導出計算

### ●$3{\alpha}$が自然数の場合

$3\displaystyle{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}{x}^{{\alpha}} ={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2} { \( x+{\Delta}x \) }^{{\alpha}}- {x}^{{\alpha}} \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}} }$

$3\displaystyle{={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}}$$3\displaystyle{ \( \hspace{1}{ }_{\alpha }\hspace{1}{C}_{0}{x}^{\alpha }\hspace{1}{+}_{\alpha }\hspace{1}{C}_{1}{x}^{ \alpha - 1 }{\Delta}x }$$3\displaystyle{\hspace{1}{+}_{\alpha }\hspace{1}{C}_{2}{x}^{ \alpha - 2 }{ \( {\Delta}x \) }^{2}+}$$3\displaystyle{\cdots \hspace{1}{+}_{\alpha }\hspace{1}{C}_{\alpha }{ \( {\Delta}x \) }^{\alpha }- {x}^{\alpha } \) }$

$3\displaystyle{={ \lim }\limits_{ {\Delta}x\rightarrow 0 } \( \hspace{1}{ }_{\alpha }\hspace{1}{C}_{1}{x}^{ \alpha - 1 }\hspace{1}{+}_{\alpha }\hspace{1}{C}_{2}{x}^{ \alpha - 2 }{\Delta}x+ }$$3\displaystyle{ \cdots \hspace{1}{+}_{\alpha }\hspace{1}{C}_{\alpha }{ \( {\Delta}x \) }^{ \alpha - 1 } \) }$

$3\displaystyle{ =\hspace{1}{}_{{\alpha}}\hspace{1}{C}_{1}{x}^{ {\alpha}- 1 } }$

$3\displaystyle{ ={\alpha}{x}^{ {\alpha}- 1 } }$

### ●$3{\alpha}$が負の整数の場合

$3\displaystyle{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}{x}^{{\alpha}} } =\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}\frac{\hspace{2}1\hspace{2}}{\hspace{2} {x}^{ - {\alpha} } \hspace{2}}$(備考：$3 - {\alpha}\text{\hspace{1} }$ は自然数となる)

$3\displaystyle{ =- \frac{\hspace{2} \frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}{x}^{ - {\alpha} } \hspace{2}}{\hspace{2} { \( {x}^{ - {\alpha} } \) }^{2} \hspace{2}} }$　　($3{}^\bullet\limits{ }_\bullet {}^\bullet$ 分数の微分)

$3\displaystyle{ =- \frac{\hspace{2} - {\alpha}{x}^{ - {\alpha}- 1 } \hspace{2}}{\hspace{2} {x}^{ - 2{\alpha} } \hspace{2}} }$

$3\displaystyle{ ={\alpha}{x}^{ - {\alpha}- 1+2{\alpha} } }$

$3\displaystyle{ ={\alpha}{x}^{ {\alpha}- 1 } }$

### ●$3 {\alpha}=0$ の場合

$3\displaystyle{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}{x}^{{\alpha}} }\displaystyle{ ={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2} { \( x+{\Delta}x \) }^{0}- {x}^{0} \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}} }$

$3\displaystyle{ ={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2}0\hspace{2}}{\hspace{2} {\Delta}x \hspace{2}} }$

=0

となり ，$3\displaystyle{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}{x}^{{\alpha}}={\alpha}{x}^{ {\alpha}- 1 } }$$3 {\alpha}=0$の場合も含む．

### ●$3{\alpha}$ が有理数の場合

$3\displaystyle{ {\alpha}=\frac{\hspace{2}p\hspace{2}}{\hspace{2}q\hspace{2}} }$ 　　$3p$ ：整数，$3p$ ：自然数として表すことができる．

$3\displaystyle{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}{x}^{{\alpha}} }\displaystyle{ =\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}{x}^{ \frac{\hspace{2}p\hspace{2}}{\hspace{2}q\hspace{2}} } }$

$3\displaystyle{ =\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}{ \( {x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}} } \) }^{p} }$

$3\displaystyle{ =p{ \( {x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}} } \) }^{ p- 1 }\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}{x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}} } }$　　(合成関数の微分)

$3\displaystyle{=p{ \( {x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}} } \) }^{ p- 1 }\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}}{x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}}- 1 }}$　　 (＊参照)

$3\displaystyle{ =\frac{\hspace{2}p\hspace{2}}{\hspace{2}q\hspace{2}}{x}^{ \frac{\hspace{2} p- 1 \hspace{2}}{\hspace{2}q\hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}}- 1 } }$

$3\displaystyle{ ={\alpha}{x}^{ {\alpha}- 1 } }$

$3\displaystyle{ x> 0 }$$3q$：自然数とする．

$3\displaystyle{ y={x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}} } }$ とすると，$3\displaystyle{ x={y}^{p} }$ となる．

$3\displaystyle{ \frac{\hspace{2} dy \hspace{2}}{\hspace{2} dx \hspace{2}} } =\frac{\hspace{2}1\hspace{2}}{\hspace{2} \frac{\hspace{2} dx \hspace{2}}{\hspace{2} dy \hspace{2}} \hspace{2}}$　　(逆関数の微分)

$3\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} q{y}^{ q- 1 } \hspace{2}} }$

$3\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}}{y}^{ 1- q } }$

$3\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}}{ \( {x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}} } \) }^{ 1- q } }$

$3\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}}{x}^{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}q\hspace{2}}- 1 } }$

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