What is (\partial)/(\partial x)(x|x|) ?

The answer to (\partial)/(\partial x)(x|x|) is (2x^2)/(|x|)

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(\partial)/(\partial x)(x|x|)

\frac{\partial}{\partial x} (x ∣ x ∣)

\frac{2 x ^{2}}{∣ x ∣}

Solution steps

\frac{\partial}{\partial x} (x ∣ x ∣)

Apply the Product Rule:

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

f = x, g = ∣ x ∣

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

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial 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 ∣}

t \to - 1 lim (\frac{e ^{t}}{( \frac{e ^{t} - 6}{5} )})

\int_{0}^{5} ∣ x - 2 ∣ d x

(1 + x)^{5}

\int_{0}^{9} \frac{1}{( x - 1 ) ^{\frac{1}{3}}} d x

f (x) = x^{3} - 2 x^{2} - 5 x + 6, at x = 2

What is (\partial)/(\partial x)(x|x|) ?
The answer to (\partial)/(\partial x)(x|x|) is (2x^2)/(|x|)