(\partial)/(\partial x)(yarcsin(x/y))

解

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

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

解答ステップ

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

y

を定数として扱う

定数を除去:

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

= y \frac{\partial}{\partial x} (arcsin (\frac{x}{y}))

連鎖法則を適用:

\frac{1}{1 - ( \frac{x}{y} ) ^{2}} \frac{\partial}{\partial x} (\frac{x}{y})

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

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

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

簡素化

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

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