derivative 5y^2(4y^2+7)^{6/5}

Lösung

ableitung von

5 y^{2} (4 y^{2} + 7)^{\frac{6}{5}}

10 y (4 y^{2} + 7)^{\frac{6}{5}} + 48 y^{3} (4 y^{2} + 7)^{\frac{1}{5}}

Schritte zur Lösung

\frac{d}{d y} (5 y^{2} (4 y^{2} + 7)^{\frac{6}{5}})

Entferne die Konstante:

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

= 5 \frac{d}{d y} (y^{2} (4 y^{2} + 7)^{\frac{6}{5}})

Wende die Produktregel an:

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

f = y^{2}, g = (4 y^{2} + 7)^{\frac{6}{5}}

= 5 (\frac{d}{d y} (y^{2}) (4 y^{2} + 7)^{\frac{6}{5}} + \frac{d}{d y} ((4 y^{2} + 7)^{\frac{6}{5}}) y^{2})

\frac{d}{d y} (y^{2}) = 2 y

\frac{d}{d y} ((4 y^{2} + 7)^{\frac{6}{5}}) = \frac{48 y ( 4 y ^{2} + 7 ) ^{\frac{1}{5}}}{5}

= 5 2 y (4 y^{2} + 7)^{\frac{6}{5}} + \frac{48 y ( 4 y ^{2} + 7 ) ^{\frac{1}{5}}}{5} y^{2}

Vereinfache

5 2 y (4 y^{2} + 7)^{\frac{6}{5}} + \frac{48 y ( 4 y ^{2} + 7 ) ^{\frac{1}{5}}}{5} y^{2} : 10 y (4 y^{2} + 7)^{\frac{6}{5}} + 48 y^{3} (4 y^{2} + 7)^{\frac{1}{5}}

= 10 y (4 y^{2} + 7)^{\frac{6}{5}} + 48 y^{3} (4 y^{2} + 7)^{\frac{1}{5}}

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