(\partial)/(\partial y)(ysin(x+y))

解

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

sin (x + y) + y cos (x + y)

解答ステップ

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

x

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 = y, g = sin (x + y)

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

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

\frac{\partial}{\partial y} (sin (x + y)) = cos (x + y)

= 1 \cdot sin (x + y) + cos (x + y) y

簡素化

= sin (x + y) + y cos (x + y)