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

解

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

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

解答ステップ

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

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

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

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

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

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

簡素化

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

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