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

解

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

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

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

商の法則を適用:

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

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

簡素化

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

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