Regra do quociente

Em matemática, a regra do quociente (ver derivada), rege a diferenciação de quocientes de funções diferenciáveis. Pode ser apresentada como: ( f g ) ′ = g f ′ − f g ′ g 2 {\displaystyle \left({\frac {f}{g}}\right)'={\frac {gf'-fg'}{g^{2}}}} ou, segundo a notação de Leibniz: d d x ( u v ) = v d u d x − u d v d x v 2 . {\displaystyle {\frac {d}{dx}}\left({\frac {u}{v}}\right)={\frac {v{\frac {du}{dx}}-u{\frac {dv}{dx}}}{v^{2}}}.} Demonstração: f ( x ) = u ( x ) v ( x ) ( I ) {\displaystyle f(x)={\frac {u(x)}{v(x)}}\ (I)} Então u ( x ) = f ( x ) ⋅ v ( x ) {\displaystyle u(x)=f(x)\cdot v(x)} Pela regra do produto: u ′ ( x ) = f ′ ( x ) ⋅ v ( x ) + f ( x ) ⋅ v ′ ( x ) ( I I ) {\displaystyle u'(x)=f'(x)\cdot v(x)+f(x)\cdot v'(x)\ (II)} Utilizando (I) e (II), temos: u ′ ( x ) = f ′ ( x ) ⋅ v ( x ) + u ( x ) v ( x ) ⋅ v ′ ( x ) {\displaystyle u'(x)=f'(x)\cdot v(x)+{\frac {u(x)}{v(x)}}\cdot v'(x)} u ′ ( x ) = f ′ ( x ) ⋅ v ( x ) ⋅ v ( x ) + u ( x ) ⋅ v ′ ( x ) v ( x ) {\displaystyle u'(x)={\frac {f'(x)\cdot v(x)\cdot v(x)+u(x)\cdot v'(x)}{v(x)}}} u ′ ( x ) ⋅ v ( x ) = f ′ ( x ) ⋅ v ( x ) ⋅ v ( x ) + u ( x ) ⋅ v ′ ( x ) {\displaystyle u'(x)\cdot v(x)=f'(x)\cdot v(x)\cdot v(x)+u(x)\cdot v'(x)} u ′ ( x ) ⋅ v ( x ) − u ( x ) ⋅ v ′ ( x )...