(\partial)/(\partial x)(x*arcsin(y/x))

解

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

arcsin (\frac{y}{x}) - \frac{y}{x ^{2} - y ^{2}}

解答ステップ

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

y

を定数として扱う

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = arcsin (\frac{y}{x})

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

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

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

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

簡素化

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

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