関数の積の微分

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

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

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

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

${\displaystyle =\underset{\begin{array}{l}\Delta x\to 0\\ c_1\to x,c_2\to x\end{array}}{\lim }\frac{\left(g^{\prime }\left(c_1\right)\Delta x\right)h\left(x\right)+g\left(x\right)\left(h^{\prime }\left(c_2\right)\Delta x\right)+\left(g^{\prime }\left(c_1\right)\Delta x\right)\left(h^{\prime }\left(c_2\right)\Delta x\right)}{\Delta x}}$

${\displaystyle =\underset{\begin{array}{l}\Delta x\to 0\\ c_1\to x,c_2\to x\end{array}}{\lim }\left\{g^{\prime }\left(c_1\right)h\left(x\right)+g\left(x\right)h^{\prime }\left(c_2\right)+g^{\prime }\left(c_1\right)h^{\prime }\left(c_2\right)\Delta x\right\}}$