derivative of (2x^x)

解

\frac{d}{d x} ((2 x)^{x})

(ln (2 x) + 1) (2 x)^{x}

解答ステップ

\frac{d}{d x} ((2 x)^{x})

指数の規則を適用する:

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

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

= \frac{d}{d x} (e^{x l n (2 x)})

連鎖法則を適用:

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

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

\frac{d}{d x} (x ln (2 x)) = ln (2 x) + 1

= e^{x l n (2 x)} (ln (2 x) + 1)

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

= (ln (2 x) + 1) (2 x)^{x}