d/(dy)((x(y))((y^6)/6+1/(16y^4))^2)

解答

\frac{d}{d y} ((x (y)) (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2})

\frac{x ( 832 y ^{20} + 144 y ^{10} - 63 )}{2304 y ^{8}}

求解步骤

\frac{d}{d y} ((x (y)) (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2})

将

x

视为常数

将常数提出:

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

= x \frac{d}{d y} (y (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2})

使用微分乘法定则:

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

f = y, g = (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2}

= x (\frac{d y}{d y} (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2} + \frac{d}{d y} ((\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2}) y)

\frac{d y}{d y} = 1

\frac{d}{d y} ((\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2}) = 2 (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}}) (y^{5} - \frac{1}{4 y ^{5}})

= x (1 \cdot (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2} + 2 (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}}) (y^{5} - \frac{1}{4 y ^{5}}) y)

化简

x (1 \cdot (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2} + 2 (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}}) (y^{5} - \frac{1}{4 y ^{5}}) y) : \frac{x ( 832 y ^{20} + 144 y ^{10} - 63 )}{2304 y ^{8}}

= \frac{x ( 832 y ^{20} + 144 y ^{10} - 63 )}{2304 y ^{8}}

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