(\partial)/(\partial x)(4(x-y)e^{4y+4x^2})

解

\frac{\partial}{\partial x} (4 (x - y) e^{4 y + 4 x^{2}})

4 (e^{4 y + 4 x^{2}} + 8 e^{4 y + 4 x^{2}} x (x - y))

解答ステップ

\frac{\partial}{\partial x} (4 (x - y) e^{4 y + 4 x^{2}})

y

を定数として扱う

定数を除去:

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

= 4 \frac{\partial}{\partial x} ((x - y) e^{4 y + 4 x^{2}})

積の法則を適用:

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

f = x - y, g = e^{4 y + 4 x^{2}}

= 4 (\frac{\partial}{\partial x} (x - y) e^{4 y + 4 x^{2}} + \frac{\partial}{\partial x} (e^{4 y + 4 x^{2}}) (x - y))

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

\frac{\partial}{\partial x} (e^{4 y + 4 x^{2}}) = e^{4 y + 4 x^{2}} \cdot 8 x

= 4 (1 \cdot e^{4 y + 4 x^{2}} + e^{4 y + 4 x^{2}} \cdot 8 x (x - y))

簡素化

= 4 (e^{4 y + 4 x^{2}} + 8 e^{4 y + 4 x^{2}} x (x - y))