derivative f(x)=-(12x)/((x^2+2)^{5/2)}

解答

导数

f (x) = - \frac{12 x}{( x ^{2} + 2 ) ^{\frac{5}{2}}}

- \frac{24 ( - 2 x ^{2} + 1 )}{( x ^{2} + 2 ) ^{\frac{7}{2}}}

求解步骤

\frac{d}{d x} (- \frac{12 x}{( x ^{2} + 2 ) ^{\frac{5}{2}}})

将常数提出:

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

= - 12 \frac{d}{d x} (\frac{x}{( x ^{2} + 2 ) ^{\frac{5}{2}}})

使用微分除法定则:

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

= - 12 \cdot \frac{\frac{d x}{d x} ( x ^{2} + 2 ) ^{\frac{5}{2}} - \frac{d}{d x} ( ( x ^{2} + 2 ) ^{\frac{5}{2}} ) x}{( ( x ^{2} + 2 ) ^{\frac{5}{2}} ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} ((x^{2} + 2)^{\frac{5}{2}}) = 5 x (x^{2} + 2)^{\frac{3}{2}}

= - 12 \cdot \frac{1 \cdot ( x ^{2} + 2 ) ^{\frac{5}{2}} - 5 x ( x ^{2} + 2 ) ^{\frac{3}{2}} x}{( ( x ^{2} + 2 ) ^{\frac{5}{2}} ) ^{2}}

化简

- 12 \cdot \frac{1 \cdot ( x ^{2} + 2 ) ^{\frac{5}{2}} - 5 x ( x ^{2} + 2 ) ^{\frac{3}{2}} x}{( ( x ^{2} + 2 ) ^{\frac{5}{2}} ) ^{2}} : - \frac{24 ( - 2 x ^{2} + 1 )}{( x ^{2} + 2 ) ^{\frac{7}{2}}}

= - \frac{24 ( - 2 x ^{2} + 1 )}{( x ^{2} + 2 ) ^{\frac{7}{2}}}

n = 1 \sum \infty \frac{1}{n !}

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