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

解答

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

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

求解步骤

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

将

x

视为常数

将常数提出:

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

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

使用指数法则:

\frac{1}{a} = a^{- 1}

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

使用连锁律:

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

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

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

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

化简

x (- \frac{1}{( x ^{2} + y ^{2} ) ^{2}} \cdot 2 y) : - \frac{2 x y}{( x ^{2} + y ^{2} ) ^{2}}

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