derivative of 100x^{0.25}y^{0.75}

Lösung

\frac{d}{d x} (100 x^{0.25} y^{0.75})

100 (\frac{0.25 y ^{0.75}}{x ^{0.75}} + \frac{0.75 x ^{0.25} \frac{d y}{d x}}{y ^{0.25}})

Schritte zur Lösung

\frac{d}{d x} (100 x^{0.25} y^{0.75})

Entferne die Konstante:

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

= 100 \frac{d}{d x} (x^{0.25} y^{0.75})

Wende die Produktregel an:

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

f = x^{0.25}, g = y^{0.75}

= 100 (\frac{d}{d x} (x^{0.25}) y^{0.75} + \frac{d}{d x} (y^{0.75}) x^{0.25})

\frac{d}{d x} (x^{0.25}) = \frac{0.25}{x ^{0.75}}

\frac{d}{d x} (y^{0.75}) = \frac{0.75 \frac{d y}{d x}}{y ^{0.25}}

= 100 (\frac{0.25}{x ^{0.75}} y^{0.75} + \frac{0.75 \frac{d y}{d x}}{y ^{0.25}} x^{0.25})

Vereinfache

100 (\frac{0.25}{x ^{0.75}} y^{0.75} + \frac{0.75 \frac{d y}{d x}}{y ^{0.25}} x^{0.25}) : 100 (\frac{0.25 y ^{0.75}}{x ^{0.75}} + \frac{0.75 x ^{0.25} \frac{d y}{d x}}{y ^{0.25}})

= 100 (\frac{0.25 y ^{0.75}}{x ^{0.75}} + \frac{0.75 x ^{0.25} \frac{d y}{d x}}{y ^{0.25}})

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