derivative f(x)=(8x^2-x+5)/(x+5)

解答

导数

f (x) = \frac{8 x ^{2} - x + 5}{x + 5}

\frac{8 x ^{2} + 80 x - 10}{( x + 5 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{8 x ^{2} - x + 5}{x + 5})

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} ( 8 x ^{2} - x + 5 ) ( x + 5 ) - \frac{d}{d x} ( x + 5 ) ( 8 x ^{2} - x + 5 )}{( x + 5 ) ^{2}}

\frac{d}{d x} (8 x^{2} - x + 5) = 16 x - 1

\frac{d}{d x} (x + 5) = 1

= \frac{( 16 x - 1 ) ( x + 5 ) - 1 \cdot ( 8 x ^{2} - x + 5 )}{( x + 5 ) ^{2}}

化简

\frac{( 16 x - 1 ) ( x + 5 ) - 1 \cdot ( 8 x ^{2} - x + 5 )}{( x + 5 ) ^{2}} : \frac{8 x ^{2} + 80 x - 10}{( x + 5 ) ^{2}}

= \frac{8 x ^{2} + 80 x - 10}{( x + 5 ) ^{2}}

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