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

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

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

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

- \frac{1}{∣ x - a ∣ ( x - a )}

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

= - \frac{1}{∣ x - a ∣ ( x - a )}

x = y^{2} - 6 y, x = 5 y - y^{2}

\int \frac{- 1}{y ^{2}} d y

\int 3 x + 4 d x

\frac{d}{d x} (\frac{a}{b})

\frac{2 x ^{6} + 1}{8 x ^{2}}

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