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Calculus · Derivatives

Differentiate x·Abs(x)

\frac{d}{dx}\left[x \left|{x}\right|\right]

Step by step

  1. \frac{d}{dx}\left[x \left|{x}\right|\right]

    Start from the derivative to compute.

  2. x \frac{d}{d x} \left|{x}\right| + \left|{x}\right| \frac{d}{d x} x

    Product rule with u = x and v = \left|{x}\right|: (uv)′ = u′v + uv′.

  3. x \operatorname{sign}{\left(x \right)} \frac{d}{d x} x + \left|{x}\right| \frac{d}{d x} x

    d/dx |u| = sign(u)·u′.

  4. x \operatorname{sign}{\left(x \right)} + \left|{x}\right|

    d/dx of x is 1.

Reveal the answer
f'(x) = x \operatorname{sign}{\left(x \right)} + \left|{x}\right|