What is the derivative of (log_{e}(x)^x) ?

The derivative of (log_{e}(x)^x) is ln^x(x)(ln(ln(x))+1/(ln(x)))

What is the first derivative of (log_{e}(x)^x) ?

The first derivative of (log_{e}(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

Popular >

\frac{d}{d x} ((lo g_{e} (x))^{x})

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

Solution steps

\frac{d}{d x} ((lo g_{e} (x))^{x})

Simplify

(lo g_{e} (x))^{x} : ln^{x} (x)

= \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 )})

y^{^{''}} + 7 y^{^{'}} + 10 y = 0, y (0) = 1, y^{^{'}} (0) = 0

x \to 0 lim (\frac{2}{3 + 4 ^{\frac{1}{x}}})

y = x^{3} + 1, (- 1, 2)

\frac{\partial}{\partial x} (y x - ln (x^{3} y + 6 x))

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