(\partial)/(\partial x)((2x^2+7x^2y)/(3xy))

解

\frac{\partial}{\partial x} (\frac{2 x ^{2} + 7 x ^{2} y}{3 x y})

\frac{7 y + 2}{3 y}

解答ステップ

\frac{\partial}{\partial x} (\frac{2 x ^{2} + 7 x ^{2} y}{3 x y})

y

を定数として扱う

定数を除去:

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

= \frac{1}{3 y} \frac{\partial}{\partial x} (\frac{2 x ^{2} + 7 x ^{2} y}{x})

商の法則を適用:

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

= \frac{1}{3 y} \cdot \frac{\frac{\partial}{\partial x} ( 2 x ^{2} + 7 x ^{2} y ) x - \frac{\partial}{\partial x} ( x ) ( 2 x ^{2} + 7 x ^{2} y )}{x ^{2}}

\frac{\partial}{\partial x} (2 x^{2} + 7 x^{2} y) = 4 x + 14 y x

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

= \frac{1}{3 y} \cdot \frac{( 4 x + 14 y x ) x - 1 \cdot ( 2 x ^{2} + 7 x ^{2} y )}{x ^{2}}

簡素化

\frac{1}{3 y} \cdot \frac{( 4 x + 14 y x ) x - 1 \cdot ( 2 x ^{2} + 7 x ^{2} y )}{x ^{2}} : \frac{7 y + 2}{3 y}

= \frac{7 y + 2}{3 y}