derivative of (12x^6-3x^2+x^2/(3x^4))

Lösung

\frac{d}{d x} (\frac{12 x ^{6} - 3 x ^{2} + x ^{2}}{3 x ^{4}})

\frac{4 ( 6 x ^{4} + 1 )}{3 x ^{3}}

Schritte zur Lösung

\frac{d}{d x} (\frac{12 x ^{6} - 3 x ^{2} + x ^{2}}{3 x ^{4}})

Entferne die Konstante:

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

= \frac{1}{3} \frac{d}{d x} (\frac{12 x ^{6} - 3 x ^{2} + x ^{2}}{x ^{4}})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{1}{3} \cdot \frac{\frac{d}{d x} ( 12 x ^{6} - 3 x ^{2} + x ^{2} ) x ^{4} - \frac{d}{d x} ( x ^{4} ) ( 12 x ^{6} - 3 x ^{2} + x ^{2} )}{( x ^{4} ) ^{2}}

\frac{d}{d x} (12 x^{6} - 3 x^{2} + x^{2}) = 72 x^{5} - 4 x

\frac{d}{d x} (x^{4}) = 4 x^{3}

= \frac{1}{3} \cdot \frac{( 72 x ^{5} - 4 x ) x ^{4} - 4 x ^{3} ( 12 x ^{6} - 3 x ^{2} + x ^{2} )}{( x ^{4} ) ^{2}}

Vereinfache

\frac{1}{3} \cdot \frac{( 72 x ^{5} - 4 x ) x ^{4} - 4 x ^{3} ( 12 x ^{6} - 3 x ^{2} + x ^{2} )}{( x ^{4} ) ^{2}} : \frac{4 ( 6 x ^{4} + 1 )}{3 x ^{3}}

= \frac{4 ( 6 x ^{4} + 1 )}{3 x ^{3}}

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