دالة المقلوب

${\displaystyle \forall h\in \mathbb {R} ^{*}\,}$

${\displaystyle {\dfrac {f(a+h)-f(a)}{h}}={\dfrac {{\dfrac {1}{a+h}}-{\dfrac {1}{a}}}{h}}}$

${\displaystyle {\dfrac {f(a+h)-f(a)}{h}}={\dfrac {\dfrac {a-(a+h)}{a(a+h)}}{h}}}$

${\displaystyle {\dfrac {f(a+h)-f(a)}{h}}={\dfrac {a-a-h}{a(a+h)}}\cdot {\dfrac {1}{h}}}$

${\displaystyle {\dfrac {f(a+h)-f(a)}{h}}={\dfrac {-1}{a(a+h)}}}$

${\displaystyle \therefore \!\,}$

${\displaystyle \lim _{h\to 0}{\dfrac {f(a+h)-f(a)}{h}}={\dfrac {-1}{a^{2}}}}$

${\displaystyle f'(a)=-{\dfrac {1}{a^{2}}}}$