derivative f(x)=(14x+4x^2)/((7+4x)^2)

解答

导数

f (x) = \frac{14 x + 4 x ^{2}}{( 7 + 4 x ) ^{2}}

\frac{98}{( 7 + 4 x ) ^{3}}

求解步骤

\frac{d}{d x} (\frac{14 x + 4 x ^{2}}{( 7 + 4 x ) ^{2}})

使用微分除法定则:

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

= \frac{\frac{d}{d x} ( 14 x + 4 x ^{2} ) ( 7 + 4 x ) ^{2} - \frac{d}{d x} ( ( 7 + 4 x ) ^{2} ) ( 14 x + 4 x ^{2} )}{( ( 7 + 4 x ) ^{2} ) ^{2}}

\frac{d}{d x} (14 x + 4 x^{2}) = 14 + 8 x

\frac{d}{d x} ((7 + 4 x)^{2}) = 8 (7 + 4 x)

= \frac{( 14 + 8 x ) ( 7 + 4 x ) ^{2} - 8 ( 7 + 4 x ) ( 14 x + 4 x ^{2} )}{( ( 7 + 4 x ) ^{2} ) ^{2}}

化简

\frac{( 14 + 8 x ) ( 7 + 4 x ) ^{2} - 8 ( 7 + 4 x ) ( 14 x + 4 x ^{2} )}{( ( 7 + 4 x ) ^{2} ) ^{2}} : \frac{98}{( 7 + 4 x ) ^{3}}

= \frac{98}{( 7 + 4 x ) ^{3}}

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