قضیه مقدار میانگین

${\displaystyle f^{\mathrm{\prime }}(c)=\frac{f(b)-f(a)}{b-a}}$

${\displaystyle \mathrm{\Phi }(a)=f(a)-\eta a=f(b)-\eta b=\mathrm{\Phi }(b)}$

${\displaystyle \eta =\frac{f(b)-f(a)}{b-a}}$

${\displaystyle \mathrm{\Phi }(x)=f(x)-\left(\frac{f(b)-f(a)}{b-a}\right)x}$

${\displaystyle {\mathrm{\Phi }}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)-\left(\frac{f(b)-f(a)}{b-a}\right)=0}$

${\displaystyle \mathrm{\Delta }f(x_0)=f(x_0+\mathrm{\Delta }x)-f(x_0)=f^{\mathrm{\prime }}(x_0+\theta \mathrm{\Delta }x)\mathrm{\Delta }x}$

${\displaystyle f^{\mathrm{\prime }}(c)=\frac{f(x_0+\mathrm{\Delta }x)-f(x_0)}{\mathrm{\Delta }x}=\frac{\mathrm{\Delta }f(x_0)}{\mathrm{\Delta }x}}$

${\displaystyle \mathrm{\Delta }f(x_0)=f^{\mathrm{\prime }}(c)\mathrm{\Delta }x(\ast )}$

${\displaystyle \mathrm{\Delta }f(x_0)=f^{\mathrm{\prime }}(x_0+\theta \mathrm{\Delta }x)\mathrm{\Delta }x}$

${\displaystyle \mathrm{\Delta }f(x_0)=(x_0+\mathrm{\Delta }x{)}^2-x_0^2=2x_0\mathrm{\Delta }x+(\mathrm{\Delta }x{)}^2}$

${\displaystyle f^{\mathrm{\prime }}(x_0+\theta x)\mathrm{\Delta }x=f^{\mathrm{\prime }}(x_0+\frac{1}{2}\mathrm{\Delta }x)\mathrm{\Delta }x}$

${\displaystyle =2(x_0+\frac{1}{2}\mathrm{\Delta }x)\mathrm{\Delta }x=2x_0\mathrm{\Delta }x+(\mathrm{\Delta }x{)}^2=\mathrm{\Delta }f(x_0)}$

${\displaystyle \frac{f^{\mathrm{\prime }}(c)}{g^{\mathrm{\prime }}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}}$