(\partial)/(\partial x)(x^2 y/(x^4+y^2))

解

\frac{\partial}{\partial x} (x^{2} \frac{y}{x ^{4} + y ^{2}})

\frac{y ( - 2 x ^{5} + 2 x y ^{2} )}{( x ^{4} + y ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (x^{2} \frac{y}{x ^{4} + y ^{2}})

y

を定数として扱う

定数を除去:

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

= y \frac{\partial}{\partial x} (x^{2} \frac{1}{x ^{4} + y ^{2}})

積の法則を適用:

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

f = x^{2}, g = \frac{1}{x ^{4} + y ^{2}}

= y (\frac{\partial}{\partial x} (x^{2}) \frac{1}{x ^{4} + y ^{2}} + \frac{\partial}{\partial x} (\frac{1}{x ^{4} + y ^{2}}) x^{2})

\frac{\partial}{\partial x} (x^{2}) = 2 x

\frac{\partial}{\partial x} (\frac{1}{x ^{4} + y ^{2}}) = - \frac{4 x ^{3}}{( x ^{4} + y ^{2} ) ^{2}}

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

簡素化

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

= \frac{y ( - 2 x ^{5} + 2 x y ^{2} )}{( x ^{4} + y ^{2} ) ^{2}}