(\partial)/(\partial x)(({f}(x,y))(3x+ye^y))

解

\frac{\partial}{\partial x} ((f (x, y)) (3 x + y e^{y}))

\frac{\partial}{\partial x} (f (x, y)) (3 x + y e^{y}) + 3 f (x, y)

解答ステップ

\frac{\partial}{\partial x} ((f (x, y)) (3 x + y e^{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 = f (x, y), g = 3 x + y e^{y}

= \frac{\partial}{\partial x} (f (x, y)) (3 x + y e^{y}) + \frac{\partial}{\partial x} (3 x + y e^{y}) f (x, y)

\frac{\partial}{\partial x} (3 x + y e^{y}) = 3

= \frac{\partial}{\partial x} (f (x, y)) (3 x + y e^{y}) + 3 f (x, y)