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

解答

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

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

求解步骤

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

将

y

视为常数

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

\frac{\partial}{\partial x} (y^{2}) = 0

= \frac{- x ^{2} + y ^{2}}{( x ^{2} + y ^{2} ) ^{2}} - 0

化简

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

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