derivative f(x)=((x^2+4)/(x^2-4))^4

Lösung

ableitung von

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

- \frac{64 x ( x ^{2} + 4 ) ^{3}}{( x ^{2} - 4 ) ^{5}}

Schritte zur Lösung

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

Wende die Kettenregel an:

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

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

\frac{d}{d x} (\frac{x ^{2} + 4}{x ^{2} - 4}) = - \frac{16 x}{( x ^{2} - 4 ) ^{2}}

= 4 (\frac{x ^{2} + 4}{x ^{2} - 4})^{3} (- \frac{16 x}{( x ^{2} - 4 ) ^{2}})

Vereinfache

4 (\frac{x ^{2} + 4}{x ^{2} - 4})^{3} (- \frac{16 x}{( x ^{2} - 4 ) ^{2}}) : - \frac{64 x ( x ^{2} + 4 ) ^{3}}{( x ^{2} - 4 ) ^{5}}

= - \frac{64 x ( x ^{2} + 4 ) ^{3}}{( x ^{2} - 4 ) ^{5}}

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