(\partial)/(\partial x)((e^{-x/2})(4x^2))

Lösung

\frac{\partial}{\partial x} ((e^{- \frac{x}{2}}) (4 x^{2}))

4 (- \frac{e ^{- \frac{x}{2}} x ^{2}}{2} + 2 e^{- \frac{x}{2}} x)

Schritte zur Lösung

\frac{\partial}{\partial x} ((e^{- \frac{x}{2}}) (4 x^{2}))

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{\partial}{\partial x} (e^{- \frac{x}{2}} x^{2})

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{- \frac{x}{2}}, g = x^{2}

= 4 (\frac{\partial}{\partial x} (e^{- \frac{x}{2}}) x^{2} + \frac{\partial}{\partial x} (x^{2}) e^{- \frac{x}{2}})

\frac{\partial}{\partial x} (e^{- \frac{x}{2}}) = - \frac{1}{2} e^{- \frac{x}{2}}

\frac{\partial}{\partial x} (x^{2}) = 2 x

= 4 ((- \frac{1}{2} e^{- \frac{x}{2}}) x^{2} + 2 x e^{- \frac{x}{2}})

Vereinfache

4 ((- \frac{1}{2} e^{- \frac{x}{2}}) x^{2} + 2 x e^{- \frac{x}{2}}) : 4 (- \frac{e ^{- \frac{x}{2}} x ^{2}}{2} + 2 e^{- \frac{x}{2}} x)

= 4 (- \frac{e ^{- \frac{x}{2}} x ^{2}}{2} + 2 e^{- \frac{x}{2}} x)

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\int_{3}^{\infty} \frac{6}{x ^{2} - x} d x

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