(\partial)/(\partial y)(e^{xy}cos(x+y))

解

\frac{\partial}{\partial y} (e^{x y} cos (x + y))

e^{x y} x cos (x + y) - e^{x y} sin (x + y)

解答ステップ

\frac{\partial}{\partial y} (e^{x y} cos (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 = e^{x y}, g = cos (x + y)

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

\frac{\partial}{\partial y} (e^{x y}) = e^{x y} x

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

= e^{x y} x cos (x + y) + (- sin (x + y)) e^{x y}

簡素化

= e^{x y} x cos (x + y) - e^{x y} sin (x + y)