derivative of (x^4/(sqrt(x^5+3)))

Lösung

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

\frac{3 x ^{8} + 24 x ^{3}}{2 ( x ^{5} + 3 ) x ^{5} + 3}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

= \frac{4 x ^{3} x ^{5} + 3 - \frac{5 x ^{4}}{2 x ^{5} + 3} x ^{4}}{( x ^{5} + 3 ) ^{2}}

Vereinfache

\frac{4 x ^{3} x ^{5} + 3 - \frac{5 x ^{4}}{2 x ^{5} + 3} x ^{4}}{( x ^{5} + 3 ) ^{2}} : \frac{3 x ^{8} + 24 x ^{3}}{2 ( x ^{5} + 3 ) x ^{5} + 3}

= \frac{3 x ^{8} + 24 x ^{3}}{2 ( x ^{5} + 3 ) x ^{5} + 3}

\frac{\partial}{\partial y} (3 x + y)

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