(\partial)/(\partial y)(8x^{5y})

解

\frac{\partial}{\partial y} (8 x^{5 y})

40 x^{5 y} ln (x)

解答ステップ

\frac{\partial}{\partial y} (8 x^{5 y})

x

を定数として扱う

定数を除去:

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

= 8 \frac{\partial}{\partial y} (x^{5 y})

指数の規則を適用する:

a^{b} = e^{b l n (a)}

x^{5 y} = e^{5 y l n (x)}

= 8 \frac{\partial}{\partial y} (e^{5 y l n (x)})

連鎖法則を適用:

e^{5 y l n (x)} \frac{\partial}{\partial y} (5 y ln (x))

= e^{5 y l n (x)} \frac{\partial}{\partial y} (5 y ln (x))

\frac{\partial}{\partial y} (5 y ln (x)) = 5 ln (x)

= 8 e^{5 y l n (x)} \cdot 5 ln (x)

簡素化

8 e^{5 y l n (x)} \cdot 5 ln (x) : 40 x^{5 y} ln (x)

= 40 x^{5 y} ln (x)