derivative of (e^x(x^5+3)/(x^{10)+9})

解答

\frac{d}{d x} (\frac{e ^{x} ( x ^{5} + 3 )}{x ^{10} + 9})

\frac{e ^{x} x ^{15} - 5 e ^{x} x ^{14} + 3 e ^{x} x ^{10} - 30 e ^{x} x ^{9} + 9 e ^{x} x ^{5} + 45 e ^{x} x ^{4} + 27 e ^{x}}{( x ^{10} + 9 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{e ^{x} ( x ^{5} + 3 )}{x ^{10} + 9})

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} ( e ^{x} ( x ^{5} + 3 ) ) ( x ^{10} + 9 ) - \frac{d}{d x} ( x ^{10} + 9 ) e ^{x} ( x ^{5} + 3 )}{( x ^{10} + 9 ) ^{2}}

\frac{d}{d x} (e^{x} (x^{5} + 3)) = e^{x} (x^{5} + 3) + 5 x^{4} e^{x}

\frac{d}{d x} (x^{10} + 9) = 10 x^{9}

= \frac{( e ^{x} ( x ^{5} + 3 ) + 5 x ^{4} e ^{x} ) ( x ^{10} + 9 ) - 10 x ^{9} e ^{x} ( x ^{5} + 3 )}{( x ^{10} + 9 ) ^{2}}

乘开

(e^{x} (x^{5} + 3) + 5 x^{4} e^{x}) (x^{10} + 9) - 10 x^{9} e^{x} (x^{5} + 3) : e^{x} x^{15} - 5 e^{x} x^{14} + 3 e^{x} x^{10} - 30 e^{x} x^{9} + 9 e^{x} x^{5} + 45 e^{x} x^{4} + 27 e^{x}

= \frac{e ^{x} x ^{15} - 5 e ^{x} x ^{14} + 3 e ^{x} x ^{10} - 30 e ^{x} x ^{9} + 9 e ^{x} x ^{5} + 45 e ^{x} x ^{4} + 27 e ^{x}}{( x ^{10} + 9 ) ^{2}}

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