derivative of (8x/(8+sqrt(x)))

Lösung

\frac{d}{d x} (\frac{8 x}{8 + x})

\frac{4 ( x + 16 )}{( 8 + x ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{8 x}{8 + x})

Entferne die Konstante:

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

= 8 \frac{d}{d x} (\frac{x}{8 + x})

Wende die Quotientenregel an:

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

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

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

\frac{d}{d x} (8 + x) = \frac{1}{2 x}

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

Vereinfache

8 \cdot \frac{1 \cdot ( 8 + x ) - \frac{1}{2 x} x}{( 8 + x ) ^{2}} : \frac{4 ( x + 16 )}{( 8 + x ) ^{2}}

= \frac{4 ( x + 16 )}{( 8 + x ) ^{2}}

\frac{d y}{d x} = (y^{2} - 9)

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