derivative of |x|^x

解

\frac{d}{d x} (∣ x ∣^{x})

∣ x ∣^{x} (ln ∣ x ∣ + 1)

解答ステップ

\frac{d}{d x} (∣ x ∣^{x})

指数の規則を適用する:

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

∣ x ∣^{x} = e^{x l n ∣ x ∣}

= \frac{d}{d x} (e^{x l n ∣ x ∣})

連鎖法則を適用:

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

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

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

= e^{x l n ∣ x ∣} (ln ∣ x ∣ + 1)

e^{x l n ∣ x ∣} = ∣ x ∣^{x}

= ∣ x ∣^{x} (ln ∣ x ∣ + 1)