derivative of (4x/(sqrt(x^2+7)))

Lösung

\frac{d}{d x} (\frac{4 x}{x ^{2} + 7})

\frac{28}{( x ^{2} + 7 ) x ^{2} + 7}

Schritte zur Lösung

\frac{d}{d x} (\frac{4 x}{x ^{2} + 7})

Entferne die Konstante:

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

= 4 \frac{d}{d x} (\frac{x}{x ^{2} + 7})

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} + 7 - \frac{d}{d x} ( x ^{2} + 7 ) x}{( x ^{2} + 7 ) ^{2}}

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

\frac{d}{d x} (x^{2} + 7) = \frac{x}{x ^{2} + 7}

= 4 \cdot \frac{1 \cdot x ^{2} + 7 - \frac{x}{x ^{2} + 7} x}{( x ^{2} + 7 ) ^{2}}

Vereinfache

4 \cdot \frac{1 \cdot x ^{2} + 7 - \frac{x}{x ^{2} + 7} x}{( x ^{2} + 7 ) ^{2}} : \frac{28}{( x ^{2} + 7 ) x ^{2} + 7}

= \frac{28}{( x ^{2} + 7 ) x ^{2} + 7}