(\partial)/(\partial x)((3x+2y)/(y^2+5x^2))

解

\frac{\partial}{\partial x} (\frac{3 x + 2 y}{y ^{2} + 5 x ^{2}})

\frac{3 y ^{2} - 15 x ^{2} - 20 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{3 x + 2 y}{y ^{2} + 5 x ^{2}})

y

を定数として扱う

商の法則を適用:

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

= \frac{\frac{\partial}{\partial x} ( 3 x + 2 y ) ( y ^{2} + 5 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 5 x ^{2} ) ( 3 x + 2 y )}{( y ^{2} + 5 x ^{2} ) ^{2}}

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

\frac{\partial}{\partial x} (y^{2} + 5 x^{2}) = 10 x

= \frac{3 ( y ^{2} + 5 x ^{2} ) - 10 x ( 3 x + 2 y )}{( y ^{2} + 5 x ^{2} ) ^{2}}

拡張

3 (y^{2} + 5 x^{2}) - 10 x (3 x + 2 y) : 3 y^{2} - 15 x^{2} - 20 x y

= \frac{3 y ^{2} - 15 x ^{2} - 20 x y}{( y ^{2} + 5 x ^{2} ) ^{2}}