derivative (1/(y^2)-5/(y^4))(y+9y^3)

解

導関数

(\frac{1}{y ^{2}} - \frac{5}{y ^{4}}) (y + 9 y^{3})

\frac{44}{y ^{2}} + \frac{15}{y ^{4}} + 9

解答ステップ

\frac{d}{d y} ((\frac{1}{y ^{2}} - \frac{5}{y ^{4}}) (y + 9 y^{3}))

積の法則を適用:

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

f = \frac{1}{y ^{2}} - \frac{5}{y ^{4}}, g = y + 9 y^{3}

= \frac{d}{d y} (\frac{1}{y ^{2}} - \frac{5}{y ^{4}}) (y + 9 y^{3}) + \frac{d}{d y} (y + 9 y^{3}) (\frac{1}{y ^{2}} - \frac{5}{y ^{4}})

\frac{d}{d y} (\frac{1}{y ^{2}} - \frac{5}{y ^{4}}) = - \frac{2}{y ^{3}} + \frac{20}{y ^{5}}

\frac{d}{d y} (y + 9 y^{3}) = 1 + 27 y^{2}

= (- \frac{2}{y ^{3}} + \frac{20}{y ^{5}}) (y + 9 y^{3}) + (1 + 27 y^{2}) (\frac{1}{y ^{2}} - \frac{5}{y ^{4}})

簡素化

(- \frac{2}{y ^{3}} + \frac{20}{y ^{5}}) (y + 9 y^{3}) + (1 + 27 y^{2}) (\frac{1}{y ^{2}} - \frac{5}{y ^{4}}) : \frac{44}{y ^{2}} + \frac{15}{y ^{4}} + 9

= \frac{44}{y ^{2}} + \frac{15}{y ^{4}} + 9