derivative of (8e^{6x}/(5x+2))

解答

\frac{d}{d x} (\frac{8 e ^{6 x}}{5 x + 2})

\frac{8 ( 30 e ^{6 x} x + 7 e ^{6 x} )}{( 5 x + 2 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{8 e ^{6 x}}{5 x + 2})

将常数提出:

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

= 8 \frac{d}{d x} (\frac{e ^{6 x}}{5 x + 2})

使用微分除法定则:

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

= 8 \cdot \frac{\frac{d}{d x} ( e ^{6 x} ) ( 5 x + 2 ) - \frac{d}{d x} ( 5 x + 2 ) e ^{6 x}}{( 5 x + 2 ) ^{2}}

\frac{d}{d x} (e^{6 x}) = e^{6 x} \cdot 6

\frac{d}{d x} (5 x + 2) = 5

= 8 \cdot \frac{e ^{6 x} \cdot 6 ( 5 x + 2 ) - 5 e ^{6 x}}{( 5 x + 2 ) ^{2}}

化简

8 \cdot \frac{e ^{6 x} \cdot 6 ( 5 x + 2 ) - 5 e ^{6 x}}{( 5 x + 2 ) ^{2}} : \frac{8 ( 30 e ^{6 x} x + 7 e ^{6 x} )}{( 5 x + 2 ) ^{2}}

= \frac{8 ( 30 e ^{6 x} x + 7 e ^{6 x} )}{( 5 x + 2 ) ^{2}}