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

解

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

\frac{3 x ^{4} - y ^{2}}{x ^{2} y}

解答ステップ

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

\frac{\partial}{\partial x} (x^{4} + y^{2}) = 4 x^{3}

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

= \frac{1}{y} \cdot \frac{4 x ^{3} x - 1 \cdot ( x ^{4} + y ^{2} )}{x ^{2}}

簡素化

\frac{1}{y} \cdot \frac{4 x ^{3} x - 1 \cdot ( x ^{4} + y ^{2} )}{x ^{2}} : \frac{3 x ^{4} - y ^{2}}{x ^{2} y}

= \frac{3 x ^{4} - y ^{2}}{x ^{2} y}