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

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

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

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

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

x^{2} ln ∣ x ∣^{2} ∣ x ∣^{∣ x ∣ - 2} + x^{2} ln ∣ x ∣ ∣ x ∣^{∣ x ∣ - 2} + ∣ x ∣^{∣ x ∣ - 1} + ln ∣ x ∣ ∣ x ∣^{∣ x ∣} + ∣ x ∣^{∣ x ∣}

Solution steps

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = ∣ x ∣^{∣ x ∣ - 1} (ln ∣ x ∣ + 1)

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

\frac{d x}{d x} = 1

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

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

Simplify

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

= x^{2} ln ∣ x ∣^{2} ∣ x ∣^{∣ x ∣ - 2} + x^{2} ln ∣ x ∣ ∣ x ∣^{∣ x ∣ - 2} + ∣ x ∣^{∣ x ∣ - 1} + ln ∣ x ∣ ∣ x ∣^{∣ x ∣} + ∣ x ∣^{∣ x ∣}

f (x) = x^{3}, at = 0

x \to 4 + lim (c x^{2} + 3 x)

\int arccot (6 y) d y

x \to 0 + lim (x^{t a n (x)})

\int ((ln (x))^{2}) d x

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