(\partial)/(\partial x)(xy*ln(x+y))

解

\frac{\partial}{\partial x} (x y \cdot ln (x + y))

y (ln (x + y) + \frac{x}{x + y})

解答ステップ

\frac{\partial}{\partial x} (x y ln (x + y))

y

を定数として扱う

定数を除去:

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

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

積の法則を適用:

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

f = x, g = ln (x + y)

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

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

\frac{\partial}{\partial x} (ln (x + y)) = \frac{1}{x + y}

= y (1 \cdot ln (x + y) + \frac{1}{x + y} x)

簡素化

y (1 \cdot ln (x + y) + \frac{1}{x + y} x) : y (ln (x + y) + \frac{x}{x + y})

= y (ln (x + y) + \frac{x}{x + y})