d/(d{x)}(({x)/y}{{x}^2+{y}^2+{z}^2})

解答

\frac{d}{d x} (\frac{x y}{x ^{2} + y ^{2} + z ^{2}})

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

求解步骤

\frac{d}{d x} (\frac{x y}{x ^{2} + y ^{2} + z ^{2}})

将常数提出:

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

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

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{d}{d x} (x) = 1

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

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

化简

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

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