derivative of-25(x/(x^2+1))

Lösung

\frac{d}{d x} (- 25 (\frac{x}{x ^{2} + 1}))

- \frac{25 ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (- 25 (\frac{x}{x ^{2} + 1}))

Vereinfache

- 25 (\frac{x}{x ^{2} + 1}) : - \frac{25 x}{x ^{2} + 1}

= \frac{d}{d x} (- \frac{25 x}{x ^{2} + 1})

Entferne die Konstante:

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

= - 25 \frac{d}{d x} (\frac{x}{x ^{2} + 1})

Wende die Quotientenregel an:

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

= - 25 \cdot \frac{\frac{d x}{d x} ( x ^{2} + 1 ) - \frac{d}{d x} ( x ^{2} + 1 ) x}{( x ^{2} + 1 ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} (x^{2} + 1) = 2 x

= - 25 \cdot \frac{1 \cdot ( x ^{2} + 1 ) - 2 xx}{( x ^{2} + 1 ) ^{2}}

Vereinfache

- 25 \cdot \frac{1 \cdot ( x ^{2} + 1 ) - 2 xx}{( x ^{2} + 1 ) ^{2}} : - \frac{25 ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{2}}

= - \frac{25 ( - x ^{2} + 1 )}{( x ^{2} + 1 ) ^{2}}

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