derivative y=(sqrt(t))^t

解

導関数

y = (t)^{t}

(t)^{t} (ln (t) + \frac{1}{2})

解答ステップ

\frac{d}{d t} ((t)^{t})

指数の規則を適用する:

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

(t)^{t} = e^{t l n (t)}

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

連鎖法則を適用:

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

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

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

= e^{t l n (t)} (ln (t) + \frac{1}{2})

e^{t l n (t)} = (t)^{t}

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