(\partial)/(\partial x)(1/4 2^xy^3)

解

\frac{\partial}{\partial x} (\frac{1}{4} 2^{x} y^{3})

ln (2) \cdot 2^{x - 2} y^{3}

解答ステップ

\frac{\partial}{\partial x} (\frac{1}{4} \cdot 2^{x} y^{3})

簡素化

\frac{1}{4} \cdot 2^{x} y^{3} : 2^{x - 2} y^{3}

= \frac{\partial}{\partial x} (2^{x - 2} y^{3})

y

を定数として扱う

定数を除去:

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

= y^{3} \frac{\partial}{\partial x} (2^{x - 2})

指数の規則を適用する:

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

2^{x - 2} = e^{(x - 2) l n (2)}

= y^{3} \frac{\partial}{\partial x} (e^{(x - 2) l n (2)})

連鎖法則を適用:

e^{(x - 2) l n (2)} \frac{\partial}{\partial x} ((x - 2) ln (2))

= e^{(x - 2) l n (2)} \frac{\partial}{\partial x} ((x - 2) ln (2))

\frac{\partial}{\partial x} ((x - 2) ln (2)) = ln (2)

= y^{3} e^{(x - 2) l n (2)} ln (2)

e^{(x - 2) l n (2)} = 2^{x - 2}

= ln (2) \cdot 2^{x - 2} y^{3}