関数の積の微分

$3$\displaystyle{{ \{ g $x$ h $x$ \} }^{\prime }}$ $3$\displaystyle{={g}^{\prime } $x$ h $x$ +g $x$ {h}^{\prime } $x$ }$

すなわち，

$3$\displaystyle{ f $x$ =g $x$ h $x$ }$ $3$\displaystyle{ \rightarrow {f}^{\prime } $x$ ={g}^{\prime } $x$ h $x$ + g $x$ h{ $x$ }^{\prime } }$

■導出

導関数の定義より

$3$\displaystyle{ {f}^{\prime } $x$ ={\lim}\limits_{ {\Delta}x \rightarrow 0 }\frac{\hspace{2} f $ x+ {\Delta}x $ - f $x$ \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}} }$

$3$\displaystyle{={ \lim }\limits_{ {\Delta}x \rightarrow 0 }\frac{\hspace{2} g\left({ x+ {\Delta}x }\right)h\left({ x+ {\Delta}x }\right)- g\left({x}\right)h\left({x}\right) \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{ {\Delta}x \rightarrow 0 }\frac{\hspace{2} g\left({ x+ {\Delta}x }\right)h\left({ x+ {\Delta}x }\right)\{\red{\text{there is tag which can't be converted to TeX}}\}- g\left({x}\right)h\left({x}\right) \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{ {\Delta}x \rightarrow 0 }\frac{\hspace{2} \left{{ g\left({ x+ {\Delta}x }\right)- g\left({x}\right) }\right}h\left({ x+ {\Delta}x }\right)+g\left({x}\right)\left{{ h\left({ x+ {\Delta}x }\right)- h\left({x}\right) }\right} \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}}$

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

$3$\displaystyle{ ={g}^{\prime } $x$ h $x$ + g $x$ {h}^{\prime } $x$ }$

よって，

$3$\displaystyle{{ \{ g $x$ h $x$ \} }^{\prime }}$ $3$\displaystyle{={g}^{\prime } $x$ h $x$ +g $x$ {h}^{\prime } $x$ }$

である．

●図形による理解

$3$\displaystyle{{\left{{ g\left({x}\right)h\left({x}\right) }\right}}^{\prime }}$ $3$\displaystyle{={ \lim }\limits_{ {\Delta}x\rightarrow 0 }\frac{\hspace{2} g\left({ x+{\Delta}x }\right)h\left({ x+{\Delta}x }\right)- g\left({x}\right)h\left({x}\right) \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{\begin{array}{lll} {\Delta}x\rightarrow 0 & \vspace{6}\\ \hspace{1}{c}_{1}\rightarrow x,\hspace{1}{c}_{2}\rightarrow x & \vspace{6}\\ \end{array}}\frac{\hspace{2} \left({ {g}^{\prime }\left({ \hspace{1}{c}_{1} }\right){\Delta}x }\right)h\left({x}\right)+g\left({x}\right)\left({ {h}^{\prime }\left({ \hspace{1}{c}_{2} }\right){\Delta}x }\right)+\left({ {g}^{\prime }\left({ \hspace{1}{c}_{1} }\right){\Delta}x }\right)\left({ {h}^{\prime }\left({ \hspace{1}{c}_{2} }\right){\Delta}x }\right) \hspace{2}}{\hspace{2} {\Delta}x \hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{\begin{array}{lll} {\Delta}x\rightarrow 0 & \vspace{6}\\ \hspace{1}{c}_{1}\rightarrow x,\hspace{1}{c}_{2}\rightarrow x & \vspace{6}\\ \end{array}}\left{{ {g}^{\prime }\left({ \hspace{1}{c}_{1} }\right)h\left({x}\right)+g\left({x}\right){h}^{\prime }\left({ \hspace{1}{c}_{2} }\right)+{g}^{\prime }\left({ \hspace{1}{c}_{1} }\right){h}^{\prime }\left({ \hspace{1}{c}_{2} }\right){\Delta}x }\right}}$

$3$\displaystyle{={g}^{\prime }\left({x}\right)h\left({x}\right)+g\left({x}\right){h}^{\prime }\left({x}\right)}$

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最終更新日： 2024年7月12日