What is the derivative of (ln(x)^x) ?

The derivative of (ln(x)^x) is ln^x(x)(ln(ln(x))+1/(ln(x)))

What is the first derivative of (ln(x)^x) ?

The first derivative of (ln(x)^x) is ln^x(x)(ln(ln(x))+1/(ln(x)))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

What is the derivative of (ln(x)^x) ?
The derivative of (ln(x)^x) is ln^x(x)(ln(ln(x))+1/(ln(x)))
What is the first derivative of (ln(x)^x) ?
The first derivative of (ln(x)^x) is ln^x(x)(ln(ln(x))+1/(ln(x)))