derivative of (6arctan(2x)/(5e^{3x)})

解答

\frac{d}{d x} (\frac{6 arctan ( 2 x )}{5 e ^{3 x}})

\frac{6 ( 2 - 3 arctan ( 2 x ) ( 1 + 4 x ^{2} ) )}{5 e ^{3 x} ( 4 x ^{2} + 1 )}

求解步骤

\frac{d}{d x} (\frac{6 arctan ( 2 x )}{5 e ^{3 x}})

将常数提出:

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

= \frac{6}{5} \frac{d}{d x} (\frac{arctan ( 2 x )}{e ^{3 x}})

使用微分除法定则:

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

= \frac{6}{5} \cdot \frac{\frac{d}{d x} ( arctan ( 2 x ) ) e ^{3 x} - \frac{d}{d x} ( e ^{3 x} ) arctan ( 2 x )}{( e ^{3 x} ) ^{2}}

\frac{d}{d x} (arctan (2 x)) = \frac{2}{4 x ^{2} + 1}

\frac{d}{d x} (e^{3 x}) = e^{3 x} \cdot 3

= \frac{6}{5} \cdot \frac{\frac{2}{4 x ^{2} + 1} e ^{3 x} - e ^{3 x} \cdot 3 arctan ( 2 x )}{( e ^{3 x} ) ^{2}}

化简

\frac{6}{5} \cdot \frac{\frac{2}{4 x ^{2} + 1} e ^{3 x} - e ^{3 x} \cdot 3 arctan ( 2 x )}{( e ^{3 x} ) ^{2}} : \frac{6 ( 2 - 3 arctan ( 2 x ) ( 1 + 4 x ^{2} ) )}{5 e ^{3 x} ( 4 x ^{2} + 1 )}

= \frac{6 ( 2 - 3 arctan ( 2 x ) ( 1 + 4 x ^{2} ) )}{5 e ^{3 x} ( 4 x ^{2} + 1 )}

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