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

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

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

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

${\displaystyle f^{\prime }(c)={\frac {f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}}={\frac {\Delta f(x_{0})}{\Delta x}}}$

${\displaystyle \Delta f(x_{0})=f^{\prime }(c)\Delta x\qquad (*)}$

${\displaystyle 0<{\frac {c-x_{0}}{\Delta x}}<1}$

${\displaystyle c=x_{0}+\theta \Delta x\qquad (0<\theta <1)}$

${\displaystyle \Delta f(x_{0})=f^{\prime }(x_{0}+\theta \Delta x)\Delta x}$

${\displaystyle \Delta f(x_{0})=(x_{0}+\Delta x)^{2}-x_{0}^{2}=2x_{0}\Delta x+(\Delta x)^{2}}$

${\displaystyle f^{\prime }(x_{0}+\theta x)\Delta x=f^{\prime }(x_{0}+{\frac {1}{2}}\Delta x)\Delta x}$

${\displaystyle =2(x_{0}+{\frac {1}{2}}\Delta x)\Delta x=2x_{0}\Delta x+(\Delta x)^{2}=\Delta f(x_{0})}$