derivative (4x)/(sqrt(x^2+2))

Lösung

ableitung von

\frac{4 x}{x ^{2} + 2}

\frac{8}{( x ^{2} + 2 ) x ^{2} + 2}

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

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

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

= 4 \cdot \frac{1 \cdot x ^{2} + 2 - \frac{x}{x ^{2} + 2} x}{( x ^{2} + 2 ) ^{2}}

Vereinfache

4 \cdot \frac{1 \cdot x ^{2} + 2 - \frac{x}{x ^{2} + 2} x}{( x ^{2} + 2 ) ^{2}} : \frac{8}{( x ^{2} + 2 ) x ^{2} + 2}

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

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