(\partial)/(\partial x)(3xarctan(x/y))

解

\frac{\partial}{\partial x} (3 x arctan (\frac{x}{y}))

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

解答ステップ

\frac{\partial}{\partial x} (3 x arctan (\frac{x}{y}))

y

を定数として扱う

定数を除去:

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

= 3 \frac{\partial}{\partial x} (x arctan (\frac{x}{y}))

積の法則を適用:

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

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

= 3 (\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}}

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

簡素化

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

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