derivative f(x)=((x^3)/(x^8+5))^2

Lösung

ableitung von

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

\frac{2 x ^{3} ( - 5 x ^{10} + 15 x ^{2} )}{( x ^{8} + 5 ) ^{3}}

Schritte zur Lösung

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

Wende die Kettenregel an:

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

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

\frac{d}{d x} (\frac{x ^{3}}{x ^{8} + 5}) = \frac{- 5 x ^{10} + 15 x ^{2}}{( x ^{8} + 5 ) ^{2}}

= 2 \cdot \frac{x ^{3}}{x ^{8} + 5} \cdot \frac{- 5 x ^{10} + 15 x ^{2}}{( x ^{8} + 5 ) ^{2}}

Vereinfache

2 \cdot \frac{x ^{3}}{x ^{8} + 5} \cdot \frac{- 5 x ^{10} + 15 x ^{2}}{( x ^{8} + 5 ) ^{2}} : \frac{2 x ^{3} ( - 5 x ^{10} + 15 x ^{2} )}{( x ^{8} + 5 ) ^{3}}

= \frac{2 x ^{3} ( - 5 x ^{10} + 15 x ^{2} )}{( x ^{8} + 5 ) ^{3}}

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