(\partial)/(\partial x)(sqrt(x/(x+y)))

解

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

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

解答ステップ

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

y

を定数として扱う

連鎖法則を適用:

\frac{1}{2 \frac{x}{x + y}} \frac{\partial}{\partial x} (\frac{x}{x + y})

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

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

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

簡素化

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

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