derivative of (d/(dx(15e^{4x}))/(8x^2))

解答

\frac{d}{d x} (\frac{\frac{d}{d x} ( 15 e ^{4 x} )}{8 x ^{2}})

\frac{15 e ^{4 x} ( 2 x ^{2} - 1 )}{x ^{4}}

求解步骤

\frac{d}{d x} (\frac{\frac{d}{d x} ( 15 e ^{4 x} )}{8 x ^{2}})

将常数提出:

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

= \frac{1}{8} \frac{d}{d x} (\frac{\frac{d}{d x} ( 15 e ^{4 x} )}{x ^{2}})

使用微分除法定则:

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

= \frac{1}{8} \cdot \frac{\frac{d}{d x} ( \frac{d}{d x} ( 15 e ^{4 x} ) ) x ^{2} - \frac{d}{d x} ( x ^{2} ) \frac{d}{d x} ( 15 e ^{4 x} )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (\frac{d}{d x} (15 e^{4 x})) = 240 e^{4 x}

\frac{d}{d x} (15 e^{4 x}) = 60 e^{4 x}

\frac{d ^{2}}{d x ^{2}} (x^{2}) = 2

\frac{d}{d x} (15 e^{4 x}) = 60 e^{4 x}

= \frac{1}{8} \cdot \frac{240 e ^{4 x} x ^{2} - 2 \cdot 60 e ^{4 x}}{( x ^{2} ) ^{2}}

化简

\frac{1}{8} \cdot \frac{240 e ^{4 x} x ^{2} - 2 \cdot 60 e ^{4 x}}{( x ^{2} ) ^{2}} : \frac{15 e ^{4 x} ( 2 x ^{2} - 1 )}{x ^{4}}

= \frac{15 e ^{4 x} ( 2 x ^{2} - 1 )}{x ^{4}}

\int 5 a^{2} x^{6} d x

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