derivative f(x)=(18x+8x^2)/((9+8x)^2)

Lösung

ableitung von

f (x) = \frac{18 x + 8 x ^{2}}{( 9 + 8 x ) ^{2}}

\frac{162}{( 9 + 8 x ) ^{3}}

Schritte zur Lösung

\frac{d}{d x} (\frac{18 x + 8 x ^{2}}{( 9 + 8 x ) ^{2}})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( 18 x + 8 x ^{2} ) ( 9 + 8 x ) ^{2} - \frac{d}{d x} ( ( 9 + 8 x ) ^{2} ) ( 18 x + 8 x ^{2} )}{( ( 9 + 8 x ) ^{2} ) ^{2}}

\frac{d}{d x} (18 x + 8 x^{2}) = 18 + 16 x

\frac{d}{d x} ((9 + 8 x)^{2}) = 16 (9 + 8 x)

= \frac{( 18 + 16 x ) ( 9 + 8 x ) ^{2} - 16 ( 9 + 8 x ) ( 18 x + 8 x ^{2} )}{( ( 9 + 8 x ) ^{2} ) ^{2}}

Vereinfache

\frac{( 18 + 16 x ) ( 9 + 8 x ) ^{2} - 16 ( 9 + 8 x ) ( 18 x + 8 x ^{2} )}{( ( 9 + 8 x ) ^{2} ) ^{2}} : \frac{162}{( 9 + 8 x ) ^{3}}

= \frac{162}{( 9 + 8 x ) ^{3}}

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