d/(dy)((1/(y^2)-7/(y^4))(y+5y^3))

解答

\frac{d}{d y} ((\frac{1}{y ^{2}} - \frac{7}{y ^{4}}) (y + 5 y^{3}))

\frac{34}{y ^{2}} + \frac{21}{y ^{4}} + 5

求解步骤

\frac{d}{d y} ((\frac{1}{y ^{2}} - \frac{7}{y ^{4}}) (y + 5 y^{3}))

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = \frac{1}{y ^{2}} - \frac{7}{y ^{4}}, g = y + 5 y^{3}

= \frac{d}{d y} (\frac{1}{y ^{2}} - \frac{7}{y ^{4}}) (y + 5 y^{3}) + \frac{d}{d y} (y + 5 y^{3}) (\frac{1}{y ^{2}} - \frac{7}{y ^{4}})

\frac{d}{d y} (\frac{1}{y ^{2}} - \frac{7}{y ^{4}}) = - \frac{2}{y ^{3}} + \frac{28}{y ^{5}}

\frac{d}{d y} (y + 5 y^{3}) = 1 + 15 y^{2}

= (- \frac{2}{y ^{3}} + \frac{28}{y ^{5}}) (y + 5 y^{3}) + (1 + 15 y^{2}) (\frac{1}{y ^{2}} - \frac{7}{y ^{4}})

化简

(- \frac{2}{y ^{3}} + \frac{28}{y ^{5}}) (y + 5 y^{3}) + (1 + 15 y^{2}) (\frac{1}{y ^{2}} - \frac{7}{y ^{4}}) : \frac{34}{y ^{2}} + \frac{21}{y ^{4}} + 5

= \frac{34}{y ^{2}} + \frac{21}{y ^{4}} + 5

in v erse l a pl a ce \frac{1}{( s ^{2} - 2 s )}

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