(\partial)/(\partial x)(x^ny 1/(x^2+y^2))

解答

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

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

求解步骤

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

将

n, y

视为常数

将常数提出:

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

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

使用微分乘法定则:

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

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

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

\frac{\partial}{\partial x} (x^{n}) = n x^{n - 1}

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

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

化简

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

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

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