Regra do quociente
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| Cálculo |
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CálculoDefinições
Conceitos |
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Cálculo integral
Definições Integração por |
| Série |
| Cálculo vetorial |
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Cálculo com múltiplas variáveisFormalismo
Definições |
| Cálculo especializado |
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 ) = f ′ ( x ) ⋅ v ( x ) ⋅ v ( x ) {\displaystyle u'(x)\cdot v(x)-u(x)\cdot v'(x)=f'(x)\cdot v(x)\cdot v(x)} u ′ ( x ) ⋅ v ( x ) − u ( x ) ⋅ v ′ ( x ) v ( x ) 2 = f ′ ( x ) {\displaystyle {\frac {u'(x)\cdot v(x)-u(x)\cdot v'(x)}{v(x)^{2}}}=f'(x)} f ′ ( x ) = u ′ v − v ′ u v 2 {\displaystyle f'(x)={\frac {u'v-v'u}{v^{2}}}}
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{\displaystyle u'(x)={\frac {f'(x)\cdot v(x)\cdot v(x)+u(x)\cdot v'(x)}{v(x)}}}
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{\displaystyle u'(x)\cdot v(x)=f'(x)\cdot v(x)\cdot v(x)+u(x)\cdot v'(x)}
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{\displaystyle u'(x)\cdot v(x)-u(x)\cdot v'(x)=f'(x)\cdot v(x)\cdot v(x)}
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{\displaystyle {\frac {u'(x)\cdot v(x)-u(x)\cdot v'(x)}{v(x)^{2}}}=f'(x)}
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{\displaystyle f'(x)={\frac {u'v-v'u}{v^{2}}}}
