derivative (1/(y^2)-3/(y^4))(y+7y^3)

解答

导数

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

\frac{20}{y ^{2}} + \frac{9}{y ^{4}} + 7

求解步骤

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

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

\frac{d}{d y} (\frac{1}{y ^{2}} - \frac{3}{y ^{4}}) = - \frac{2}{y ^{3}} + \frac{12}{y ^{5}}

\frac{d}{d y} (y + 7 y^{3}) = 1 + 21 y^{2}

= (- \frac{2}{y ^{3}} + \frac{12}{y ^{5}}) (y + 7 y^{3}) + (1 + 21 y^{2}) (\frac{1}{y ^{2}} - \frac{3}{y ^{4}})

化简

(- \frac{2}{y ^{3}} + \frac{12}{y ^{5}}) (y + 7 y^{3}) + (1 + 21 y^{2}) (\frac{1}{y ^{2}} - \frac{3}{y ^{4}}) : \frac{20}{y ^{2}} + \frac{9}{y ^{4}} + 7

= \frac{20}{y ^{2}} + \frac{9}{y ^{4}} + 7

\frac{\partial}{\partial x} (2 x e^{2 x y})

x \to + 7 lim (\frac{x}{x + 7})

\frac{\partial}{\partial x} (\frac{x e ^{y}}{3 z ^{2} + 1})

d er i v a t i v e 3 t^{2} - 5 t^{3}

\frac{d y}{d x} + 3 x y^{3} = 0, y (0) = 1