derivative of (5x^6+4x^5+3x^4/(6x^2))

Lösung

\frac{d}{d x} (\frac{5 x ^{6} + 4 x ^{5} + 3 x ^{4}}{6 x ^{2}})

\frac{x ( 10 x ^{2} + 6 x + 3 )}{3}

Schritte zur Lösung

\frac{d}{d x} (\frac{5 x ^{6} + 4 x ^{5} + 3 x ^{4}}{6 x ^{2}})

Entferne die Konstante:

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

= \frac{1}{6} \frac{d}{d x} (\frac{5 x ^{6} + 4 x ^{5} + 3 x ^{4}}{x ^{2}})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (5 x^{6} + 4 x^{5} + 3 x^{4}) = 30 x^{5} + 20 x^{4} + 12 x^{3}

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

= \frac{1}{6} \cdot \frac{( 30 x ^{5} + 20 x ^{4} + 12 x ^{3} ) x ^{2} - 2 x ( 5 x ^{6} + 4 x ^{5} + 3 x ^{4} )}{( x ^{2} ) ^{2}}

Vereinfache

\frac{1}{6} \cdot \frac{( 30 x ^{5} + 20 x ^{4} + 12 x ^{3} ) x ^{2} - 2 x ( 5 x ^{6} + 4 x ^{5} + 3 x ^{4} )}{( x ^{2} ) ^{2}} : \frac{x ( 10 x ^{2} + 6 x + 3 )}{3}

= \frac{x ( 10 x ^{2} + 6 x + 3 )}{3}

d er i v a t i v e \frac{x ^{2} - 2 e ^{x}}{x + 3 e ^{x}}

\frac{d}{d x} (\frac{x ^{3} + 1}{x ^{3} - 1})

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