derivative of (x-1^{2x})

解

\frac{d}{d x} ((x - 1)^{2 x})

2 (ln (x - 1) + \frac{x}{x - 1}) (x - 1)^{2 x}

解答ステップ

\frac{d}{d x} ((x - 1)^{2 x})

指数の規則を適用する:

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

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

= \frac{d}{d x} (e^{2 x l n (x - 1)})

連鎖法則を適用:

e^{2 x l n (x - 1)} \frac{d}{d x} (2 x ln (x - 1))

= e^{2 x l n (x - 1)} \frac{d}{d x} (2 x ln (x - 1))

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

= e^{2 x l n (x - 1)} \cdot 2 (ln (x - 1) + \frac{x}{x - 1})

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

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