derivative y=((x^2+3)/(x^2-3))^4

Lösung

ableitung von

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

- \frac{48 x ( x ^{2} + 3 ) ^{3}}{( x ^{2} - 3 ) ^{5}}

Schritte zur Lösung

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

4 (\frac{x ^{2} + 3}{x ^{2} - 3})^{3} (- \frac{12 x}{( x ^{2} - 3 ) ^{2}}) : - \frac{48 x ( x ^{2} + 3 ) ^{3}}{( x ^{2} - 3 ) ^{5}}

= - \frac{48 x ( x ^{2} + 3 ) ^{3}}{( x ^{2} - 3 ) ^{5}}

x \to 1 lim (- x)

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