(\partial)/(\partial x)(-u(x,y)cos(kx)sin(ky))

解

\frac{\partial}{\partial x} (- u (x, y) cos (k x) sin (k y))

- sin (k y) (\frac{\partial}{\partial x} (u (x, y)) cos (k x) + (- sin (k x) k) u (x, y))

解答ステップ

\frac{\partial}{\partial x} (- u (x, y) cos (k x) sin (k y))

k, y

を定数として扱う

定数を除去:

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

= - sin (k y) \frac{\partial}{\partial x} (u (x, y) cos (k x))

積の法則を適用:

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

f = u (x, y), g = cos (k x)

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

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

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