積の差分の証明

$\frac{{∆}_h(f\left(x\right)g\left(x\right))}{{∆}_{h}x}=\frac{f(x+h)g(x+h)-f\left(x\right)g\left(x\right)}{(x+h)-x}$
$=\frac{f(x+h)g(x+h)-f\left(x\right)g\left(x\right)}{h}+\frac{f\left(x\right)g(x+h)-f\left(x\right)g(x+h)}{h}$
$=\frac{f(x+h)g(x+h)-f\left(x\right)g(x+h)}{h}+\frac{f\left(x\right)g(x+h)-f\left(x\right)g\left(x\right)}{h}$
$=\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g(x+h)+f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}$
$=\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g(x+h)+f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+(\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g\left(x\right)-\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g\left(x\right))$
$=f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g\left(x\right)+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}(g(x+h)-g\left(x\right))$
$=f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g\left(x\right)+h\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}$
$\text{よって}$
$\frac{{∆}_h(f\left(x\right)g\left(x\right))}{{∆}_{h}x}=f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g\left(x\right)+h\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}$

$\frac{{\nabla }_h(f\left(x\right)g\left(x\right))}{{\nabla }_{h}x}=\frac{f\left(x\right)g\left(x\right)-f(x-h)g(x-h)}{h}$
$=\frac{f\left(x\right)g\left(x\right)-f(x-h)g(x-h)}{h}+\frac{f\left(x\right)g(x-h)-f\left(x\right)g(x-h)}{h}$
$=\frac{f\left(x\right)g\left(x\right)-f\left(x\right)g(x-h)}{h}+\frac{f\left(x\right)g(x-h)-f(x-h)g(x-h)}{h}$
$=f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g(x-h)$
$=f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g(x-h)+(\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g\left(x\right))$
$=f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g\left(x\right)+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}(g(x-h)-g\left(x\right))$
$=f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-h\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}$

$\text{よって}$
$\frac{{\nabla }_h(f\left(x\right)g\left(x\right))}{{\nabla }_{h}x}=f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-h\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}$

$M_{x}f\left(x\right)=\frac{f(x+\frac{h}{2})+f(x-\frac{h}{2})}{2}$
$=\frac{2f(x+\frac{h}{2})-f(x+\frac{h}{2})+f(x-\frac{h}{2})}{2}$
$=f(x+\frac{h}{2})-\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}$
$\text{変形して}$
$f(x+\frac{h}{2})=M_{x}f\left(x\right)+\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}\text{…①}$

$M_{x}f\left(x\right)=\frac{f(x+\frac{h}{2})+f(x-\frac{h}{2})}{2}$
$=\frac{f(x+\frac{h}{2})-f(x-\frac{h}{2})+2f(x-\frac{h}{2})}{2}$
$=f(x-\frac{h}{2})+\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}$
$\text{変形して}$
$f(x-\frac{h}{2})=M_{x}f\left(x\right)-\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}\text{…②}$

$M_{x}g\left(x\right)=\frac{g(x+\frac{h}{2})+g(x-\frac{h}{2})}{2}$
$=\frac{2g(x+\frac{h}{2})-g(x+\frac{h}{2})+g(x-\frac{h}{2})}{2}$
$=g(x+\frac{h}{2})-\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}$
$\text{変形して}$
$g(x+\frac{h}{2})=M_{x}g\left(x\right)+\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}\text{…③}$

$M_{x}g\left(x\right)=\frac{g(x+\frac{h}{2})+g(x-\frac{h}{2})}{2}$
$=\frac{g(x+\frac{h}{2})-g(x-\frac{h}{2})+2g(x-\frac{h}{2})}{2}$
$=g(x-\frac{h}{2})+\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}$
$\text{変形して}$
$g(x-\frac{h}{2})=M_{x}g\left(x\right)-\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}\text{…④}$

$\text{①～④より}$
$\frac{{\delta }_h(f\left(x\right)g\left(x\right))}{{\delta }_{h}x}=\frac{f(x+\frac{h}{2})g(x+\frac{h}{2})-f(x-\frac{h}{2})g(x-\frac{h}{2})}{h}$
$=\frac{(M_{x}f\left(x\right)+\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x})(M_{x}g\left(x\right)+\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x})}{h}-\frac{(M_{x}f\left(x\right)-\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x})(M_{x}g\left(x\right)-\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x})}{h}$
$=\frac{M_{x}f\left(x\right)M_{x}g\left(x\right)+\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)+M_{x}f\left(x\right)\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}+{\left(\frac{h}{2}\right)}^2\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}}{h}-\frac{M_{x}f\left(x\right)M_{x}g\left(x\right)-\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)-M_{x}f\left(x\right)\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}+{\left(\frac{h}{2}\right)}^2\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}}{h}$
$=\frac{\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)+M_{x}f\left(x\right)\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}}{h}-\frac{-\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)-M_{x}f\left(x\right)\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}}{h}$
$=\frac{\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)+M_{x}f\left(x\right)\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}}{h}+\frac{\frac{h}{2}\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)+M_{x}f\left(x\right)\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}}{h}$
$=\frac{h\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)+M_{x}f\left(x\right)h\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}}{h}$
$=\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)+M_{x}f\left(x\right)\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}$

$\text{よって}$
$\frac{{\delta }_h(f\left(x\right)g\left(x\right))}{{\delta }_{h}x}=M_{x}f\left(x\right)\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}+\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)$

$\frac{{∆}_h(f\left(x\right)g\left(x\right))}{{∆}_{h}x}=f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g\left(x\right)+h\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}$
$\frac{{\nabla }_h(f\left(x\right)g\left(x\right))}{{\nabla }_{h}x}=f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-h\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}$
$\frac{{\delta }_h(f\left(x\right)g\left(x\right))}{{\delta }_{h}x}=M_{x}f\left(x\right)\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x}+\frac{{\delta }_{h}f\left(x\right)}{{\delta }_{h}x}{M}_{x}g\left(x\right)$
$\text{ただし}$
$M_{x}f\left(x\right)=\frac{f(x+\frac{h}{2})+f(x-\frac{h}{2})}{2}$
$M_{x}g\left(x\right)=\frac{g(x+\frac{h}{2})+g(x-\frac{h}{2})}{2}$