What is (\partial)/(\partial x)(1/(|x|)) ?

The answer to (\partial)/(\partial x)(1/(|x|)) is -1/(x|x|)

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(\partial)/(\partial x)(1/(|x|))

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

- \frac{1}{x ∣ x ∣}

Solution steps

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= \frac{\partial}{\partial x} (∣ x ∣^{- 1})

Apply the chain rule:

- \frac{1}{∣ x ∣ ^{2}} \frac{\partial}{\partial x} (∣ x ∣)

= - \frac{1}{∣ x ∣ ^{2}} \frac{\partial}{\partial x} (∣ x ∣)

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

= - \frac{1}{∣ x ∣ ^{2}} \cdot \frac{x}{∣ x ∣}

Simplify

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

= - \frac{1}{x ∣ x ∣}

\int_{0}^{2} 2 x + 5 d x

\frac{d}{d x} (40 x (5 x^{2} - 8))

\frac{\partial}{\partial y} (8 x y)

\int x^{- \frac{1}{3}} d x

\frac{\partial}{\partial x} (2 e^{2} + e^{y} (2 x + 2 y))

What is (\partial)/(\partial x)(1/(|x|)) ?
The answer to (\partial)/(\partial x)(1/(|x|)) is -1/(x|x|)