(\partial)/(\partial x)(K(x)^{1/3}L^{2/3})

Lösung

\frac{\partial}{\partial x} (K (x) (x)^{\frac{1}{3}} L^{\frac{2}{3}})

\frac{L ^{\frac{2}{3}} ( K ( x ) + 3 x \frac{\partial}{\partial x} ( K ( x ) ) )}{3 x ^{\frac{2}{3}}}

Schritte zur Lösung

\frac{\partial}{\partial x} (K (x) (x)^{\frac{1}{3}} L^{\frac{2}{3}})

Entferne die Konstante:

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

= L^{\frac{2}{3}} \frac{\partial}{\partial x} (K (x) x^{\frac{1}{3}})

Wende die Produktregel an:

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

f = K (x), g = x^{\frac{1}{3}}

= L^{\frac{2}{3}} (\frac{\partial}{\partial x} (K (x)) x^{\frac{1}{3}} + \frac{\partial}{\partial x} (x^{\frac{1}{3}}) K (x))

\frac{\partial}{\partial x} (x^{\frac{1}{3}}) = \frac{1}{3 x ^{\frac{2}{3}}}

= L^{\frac{2}{3}} (\frac{\partial}{\partial x} (K (x)) x^{\frac{1}{3}} + \frac{1}{3 x ^{\frac{2}{3}}} K (x))

Vereinfache

L^{\frac{2}{3}} (\frac{\partial}{\partial x} (K (x)) x^{\frac{1}{3}} + \frac{1}{3 x ^{\frac{2}{3}}} K (x)) : \frac{L ^{\frac{2}{3}} ( K ( x ) + 3 x \frac{\partial}{\partial x} ( K ( x ) ) )}{3 x ^{\frac{2}{3}}}

= \frac{L ^{\frac{2}{3}} ( K ( x ) + 3 x \frac{\partial}{\partial x} ( K ( x ) ) )}{3 x ^{\frac{2}{3}}}

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