# 関数の積の微分

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

すなわち，

$f\left(x\right)=g\left(x\right)h\left(x\right)$$\to {f}^{\prime }\left(x\right)={g}^{\prime }\left(x\right)h\left(x\right)+g\left(x\right)h{\left(x\right)}^{\prime }$

## ■導出

${f}^{\prime }\left(x\right)=\underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\Delta x}$

$=\underset{\Delta x\to 0}{\mathrm{lim}}\frac{g\left(x+\Delta x\right)h\left(x+\Delta x\right)-g\left(x\right)h\left(x\right)}{\Delta x}$

$=\underset{\Delta x\to 0}{\mathrm{lim}}\frac{g\left(x+\Delta x\right)h\left(x+\Delta x\right)-g\left(x\right)h\left(x+\Delta x\right)+g\left(x\right)h\left(x+\Delta x\right)-g\left(x\right)h\left(x\right)}{\Delta x}$

$=\underset{\Delta x\to 0}{\mathrm{lim}}\frac{\left\{g\left(x+\Delta x\right)-g\left(x\right)\right\}h\left(x+\Delta x\right)+g\left(x\right)\left\{h\left(x+\Delta x\right)-h\left(x\right)\right\}}{\Delta x}$

$=\left\{\underset{\Delta x\to 0}{lim}\frac{g\left(x+\Delta x\right)-g\left(x\right)}{\Delta x}\right\}$$\left\{\underset{\Delta x\to 0}{lim}h\left(x+\Delta x\right)\right\}$$+g\left(x\right)\left\{\underset{\Delta x\to 0}{lim}\frac{h\left(x+\Delta x\right)-h\left(x\right)}{\Delta x}\right\}$

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

よって，

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

である．

### ●図形による理解

${\left\{g\left(x\right)h\left(x\right)\right\}}^{\prime }$ $=\underset{\Delta x\to 0}{\mathrm{lim}}\frac{g\left(x+\Delta x\right)h\left(x+\Delta x\right)-g\left(x\right)h\left(x\right)}{\Delta x}$

$=\underset{\begin{array}{l}\Delta x\to 0\\ {c}_{1}\to x,{c}_{2}\to x\end{array}}{\mathrm{lim}}\frac{\left({g}^{\prime }\left({c}_{1}\right)\Delta x\right)h\left(x\right)+g\left(x\right)\left({h}^{\prime }\left({c}_{2}\right)\Delta x\right)+\left({g}^{\prime }\left({c}_{1}\right)\Delta x\right)\left({h}^{\prime }\left({c}_{2}\right)\Delta x\right)}{\Delta x}$

$=\underset{\begin{array}{l}\Delta x\to 0\\ {c}_{1}\to x,{c}_{2}\to x\end{array}}{\mathrm{lim}}\left\{{g}^{\prime }\left({c}_{1}\right)h\left(x\right)+g\left(x\right){h}^{\prime }\left({c}_{2}\right)+{g}^{\prime }\left({c}_{1}\right){h}^{\prime }\left({c}_{2}\right)\Delta x\right\}$

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

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