(\partial)/(\partial x)(xarctan(x+2y))

解

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

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

解答ステップ

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

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

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

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

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

簡素化

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

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