商の差分の証明

$\frac{{∆}_h\frac{f\left(x\right)}{g\left(x\right)}}{{∆}_{h}x}=\frac{{∆}_h(f\left(x\right)\frac{1}{g\left(x\right)})}{{∆}_{h}x}$
$=f\left(x\right)\frac{{∆}_h\frac{1}{g\left(x\right)}}{{∆}_{h}x}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{1}{g\left(x\right)}+h\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{{∆}_h\frac{1}{g\left(x\right)}}{{∆}_{h}x}$
$=f\left(x\right)\frac{\left(\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}\right)}{{\left(g\left(x\right)\right)}^2-h\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}g\left(x\right)}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{1}{g\left(x\right)}+h\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{\left(\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}\right)}{{\left(g\left(x\right)\right)}^2-h\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}g\left(x\right)}$
$=\frac{f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}(g\left(x\right)-h\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x})+h\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}}{{\left(g\left(x\right)\right)}^2-h\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}g\left(x\right)}$
$=\frac{f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g\left(x\right)-h\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+h\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}}{{\left(g\left(x\right)\right)}^2-h\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}g\left(x\right)}$
$=\frac{f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g\left(x\right)}{{\left(g\left(x\right)\right)}^2-h\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}g\left(x\right)}$
$\text{よって}$
$\frac{{∆}_h\frac{f\left(x\right)}{g\left(x\right)}}{{∆}_{h}x}=\frac{f\left(x\right)\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}+\frac{{∆}_{h}f\left(x\right)}{{∆}_{h}x}g\left(x\right)}{{\left(g\left(x\right)\right)}^2-h\frac{{∆}_{h}g\left(x\right)}{{∆}_{h}x}g\left(x\right)}$

$\frac{{\nabla }_h\frac{f\left(x\right)}{g\left(x\right)}}{{\nabla }_{h}x}=\frac{{\nabla }_h(f\left(x\right)\frac{1}{g\left(x\right)})}{{\nabla }_{h}x}$
$=f\left(x\right)\frac{{\nabla }_h\frac{1}{g\left(x\right)}}{{\nabla }_{h}x}+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{1}{g\left(x\right)}-h\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{{\nabla }_h\frac{1}{g\left(x\right)}}{{\nabla }_{h}x}$
$=f\left(x\right)\frac{\left(\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}\right)}{h\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-{\left(g\left(x\right)\right)}^2}+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{1}{g\left(x\right)}-h\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{\left(\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}\right)}{h\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-{\left(g\left(x\right)\right)}^2}$
$=\frac{f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}+\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}(h\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}-g\left(x\right))-h\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}}{h\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-{\left(g\left(x\right)\right)}^2}$
$=\frac{f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}-\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g\left(x\right)+h\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}-h\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}}{h\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-{\left(g\left(x\right)\right)}^2}$
$=\frac{f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}-\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g\left(x\right)}{h\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-{\left(g\left(x\right)\right)}^2}$
$\text{よって}$
$\frac{{\nabla }_h\frac{f\left(x\right)}{g\left(x\right)}}{{\nabla }_{h}x}=\frac{f\left(x\right)\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}-\frac{{\nabla }_{h}f\left(x\right)}{{\nabla }_{h}x}g\left(x\right)}{h\frac{{\nabla }_{h}g\left(x\right)}{{\nabla }_{h}x}g\left(x\right)-{\left(g\left(x\right)\right)}^2}$

$M_x\frac{1}{g\left(x\right)}=\frac{\frac{1}{g(x+\frac{h}{2})}+\frac{1}{g(x-\frac{h}{2})}}{2}$
$=\frac{g(x-\frac{h}{2})+g(x+\frac{h}{2})}{2g(x+\frac{h}{2})g(x-\frac{h}{2})}$
$=\frac{M_{x}g\left(x\right)}{g(x+\frac{h}{2})g(x-\frac{h}{2})}$
$=\frac{M_{x}g\left(x\right)}{{\left(M_{x}g\left(x\right)\right)}^2-{(\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x})}^2}$
$=-\frac{M_{x}g\left(x\right)}{{(\frac{h}{2}\frac{{\delta }_{h}g\left(x\right)}{{\delta }_{h}x})}^2-{\left(M_{x}g\left(x\right)\right)}^2}\text{…①}$

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

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