derivative of x^{15x}

解

\frac{d}{d x} (x^{15 x})

15 x^{15 x} (ln (x) + 1)

解答ステップ

\frac{d}{d x} (x^{15 x})

指数の規則を適用する:

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

x^{15 x} = e^{15 x l n (x)}

= \frac{d}{d x} (e^{15 x l n (x)})

連鎖法則を適用:

e^{15 x l n (x)} \frac{d}{d x} (15 x ln (x))

= e^{15 x l n (x)} \frac{d}{d x} (15 x ln (x))

\frac{d}{d x} (15 x ln (x)) = 15 (ln (x) + 1)

= e^{15 x l n (x)} \cdot 15 (ln (x) + 1)

e^{15 x l n (x)} = x^{15 x}

= 15 x^{15 x} (ln (x) + 1)