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

解

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

\frac{x ^{2} - y ^{2} + 4}{x ^{2} y}

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

\frac{1}{y} \cdot \frac{2 xx - 1 \cdot ( x ^{2} + y ^{2} - 4 )}{x ^{2}} : \frac{x ^{2} - y ^{2} + 4}{x ^{2} y}

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