(\partial)/(\partial x)(0.5xarctan(y/x))

解

\frac{\partial}{\partial x} (0.5 x arctan (\frac{y}{x}))

0.5 (arctan (\frac{y}{x}) - \frac{x y}{y ^{2} + x ^{2}})

解答ステップ

\frac{\partial}{\partial x} (0.5 x arctan (\frac{y}{x}))

y

を定数として扱う

定数を除去:

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

= 0.5 \frac{\partial}{\partial x} (x arctan (\frac{y}{x}))

積の法則を適用:

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

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

= 0.5 (\frac{\partial}{\partial x} (x) arctan (\frac{y}{x}) + \frac{\partial}{\partial x} (arctan (\frac{y}{x})) x)

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

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

= 0.5 (1 \cdot arctan (\frac{y}{x}) + (- \frac{y}{y ^{2} + x ^{2}}) x)

簡素化

0.5 (1 \cdot arctan (\frac{y}{x}) + (- \frac{y}{y ^{2} + x ^{2}}) x) : 0.5 (arctan (\frac{y}{x}) - \frac{x y}{y ^{2} + x ^{2}})

= 0.5 (arctan (\frac{y}{x}) - \frac{x y}{y ^{2} + x ^{2}})