What is the derivative of x|x| ?

The derivative of x|x| is (2x^2)/(|x|)

What is the first derivative of x|x| ?

The first derivative of x|x| is (2x^2)/(|x|)

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derivative

\frac{2 x ^{2}}{∣ x ∣}

Solution steps

\frac{d}{d x} (x ∣ x ∣)

Apply the Product Rule:

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

f = x, g = ∣ x ∣

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

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

\frac{d}{d x} (∣ x ∣) = \frac{x}{∣ x ∣}

= 1 \cdot ∣ x ∣ + \frac{x}{∣ x ∣} x

Simplify

1 \cdot ∣ x ∣ + \frac{x}{∣ x ∣} x : \frac{2 x ^{2}}{∣ x ∣}

= \frac{2 x ^{2}}{∣ x ∣}

What is the derivative of x|x| ?
The derivative of x|x| is (2x^2)/(|x|)
What is the first derivative of x|x| ?
The first derivative of x|x| is (2x^2)/(|x|)