משפט

אם ${\displaystyle f,g}$ גזירות ומתקיים ${\displaystyle g'\neq 0}$ , אזי ${\displaystyle {\frac {f}{g}}}$ גזירה ומתקיים ${\displaystyle {\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}}$ .

$${\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {{\dfrac {f(x+h)}{g(x+h)}}-{\dfrac {f(x)}{g(x)}}}{h}}&=\lim _{h\to 0}{\dfrac {\dfrac {f(x+h)g(x)-f(x)g(x+h)}{h}}{g(x)g(x+h)}}\\&=\lim _{h\to 0}{\dfrac {\dfrac {f(x+h)g(x)-{\color {blue}f(x)g(x)}-f(x)g(x+h)+{\color {blue}f(x)g(x)}}{h}}{g(x)g(x+h)}}\\&=\lim _{h\to 0}{\dfrac {g(x)\cdot {\dfrac {f(x+h)-f(x)}{h}}-f(x)\cdot {\dfrac {g(x+h)-g(x)}{h}}}{g(x)g(x+h)}}\\&={\dfrac {\lim \limits _{h\to 0}\left[g(x)\cdot {\dfrac {f(x+h)-f(x)}{h}}-f(x)\cdot {\dfrac {g(x+h)-g(x)}{h}}\right]}{\lim \limits _{h\to 0}g(x)g(x+h)}}\\&={\dfrac {\lim \limits _{h\to 0}g(x)\cdot {\dfrac {f(x+h)-f(x)}{h}}-\lim \limits _{h\to 0}f(x)\cdot {\dfrac {g(x+h)-g(x)}{h}}}{\lim \limits _{h\to 0}g(x)g(x+h)}}\\&={\dfrac {\lim \limits _{h\to 0}g(x)\cdot \lim \limits _{h\to 0}{\dfrac {f(x+h)-f(x)}{h}}-\lim \limits _{h\to 0}f(x)\cdot \lim \limits _{h\to 0}{\dfrac {g(x+h)-g(x)}{h}}}{\lim \limits _{h\to 0}g(x)\cdot \lim \limits _{h\to 0}g(x+h)}}\\&={\dfrac {g(x)\cdot \lim \limits _{h\to 0}{\dfrac {f(x+h)-f(x)}{h}}-f(x)\cdot \lim \limits _{h\to 0}{\dfrac {g(x+h)-g(x)}{h}}}{g(x)^{2}}}\\&={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}\end{aligned}}}$$

${\displaystyle \blacksquare }$

נגדיר ${\displaystyle h(x)={\frac {f(x)}{g(x)}}}$ ומכאן ${\displaystyle f(x)=g(x)h(x)}$ . נשתמש בכלל למכפלת נגזרות ונקבל:

$${\displaystyle {\begin{matrix}f'(x)=g'(x)h(x)+g(x)h'(x)\\h'(x)={\dfrac {f'(x)-g'(x)h(x)}{g(x)}}={\dfrac {f'(x)-g'(x)\cdot {\dfrac {f(x)}{g(x)}}}{g(x)}}={\dfrac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}\end{matrix}}}$$

${\displaystyle \blacksquare }$