(\partial)/(\partial x)(2(x-y)e^{2y+5x^2})

解

\frac{\partial}{\partial x} (2 (x - y) e^{2 y + 5 x^{2}})

2 (e^{2 y + 5 x^{2}} + 10 e^{2 y + 5 x^{2}} x (x - y))

解答ステップ

\frac{\partial}{\partial x} (2 (x - y) e^{2 y + 5 x^{2}})

y

を定数として扱う

定数を除去:

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

= 2 \frac{\partial}{\partial x} ((x - y) e^{2 y + 5 x^{2}})

積の法則を適用:

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

f = x - y, g = e^{2 y + 5 x^{2}}

= 2 (\frac{\partial}{\partial x} (x - y) e^{2 y + 5 x^{2}} + \frac{\partial}{\partial x} (e^{2 y + 5 x^{2}}) (x - y))

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

\frac{\partial}{\partial x} (e^{2 y + 5 x^{2}}) = e^{2 y + 5 x^{2}} \cdot 10 x

= 2 (1 \cdot e^{2 y + 5 x^{2}} + e^{2 y + 5 x^{2}} \cdot 10 x (x - y))

簡素化

= 2 (e^{2 y + 5 x^{2}} + 10 e^{2 y + 5 x^{2}} x (x - y))