What is the derivative of |x|^{|x|} ?

The derivative of |x|^{|x|} is x|x|^{|x|-1}(ln|x|+1)

What is the first derivative of |x|^{|x|} ?

The first derivative of |x|^{|x|} is x|x|^{|x|-1}(ln|x|+1)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

\frac{d}{d x} (∣ x ∣^{∣ x ∣})

x ∣ x ∣^{∣ x ∣ - 1} (ln ∣ x ∣ + 1)

Solution steps

\frac{d}{d x} (∣ x ∣^{∣ x ∣})

Apply exponent rule:

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

∣ x ∣^{∣ x ∣} = e^{∣ x ∣ l n ∣ x ∣}

= \frac{d}{d x} (e^{∣ x ∣ l n ∣ x ∣})

Apply the chain rule:

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

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

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

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

Simplify

e^{∣ x ∣ l n ∣ x ∣} \frac{x ( ln ∣ x ∣ + 1 )}{∣ x ∣} : x ∣ x ∣^{∣ x ∣ - 1} (ln ∣ x ∣ + 1)

= x ∣ x ∣^{∣ x ∣ - 1} (ln ∣ x ∣ + 1)

What is the derivative of |x|^{|x|} ?
The derivative of |x|^{|x|} is x|x|^{|x|-1}(ln|x|+1)
What is the first derivative of |x|^{|x|} ?
The first derivative of |x|^{|x|} is x|x|^{|x|-1}(ln|x|+1)