What is (\partial)/(\partial x)(In|x-y|) ?

The answer to (\partial)/(\partial x)(In|x-y|) is (nI(x-y))/(|x-y|)

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(\partial)/(\partial x)(In|x-y|)

\frac{\partial}{\partial x} (I n ∣ x - y ∣)

\frac{n I ( x - y )}{∣ x - y ∣}

Solution steps

\frac{\partial}{\partial x} (I n ∣ x - y ∣)

Treat

n, I, y

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= I n \frac{\partial}{\partial x} (∣ x - y ∣)

Apply the chain rule:

\frac{x - y}{∣ ( x - y ) ∣} \frac{\partial}{\partial x} (x - y)

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

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

= I n \frac{x - y}{∣ ( x - y ) ∣} \cdot 1

Simplify

I n \frac{x - y}{∣ x - y ∣} \cdot 1 : \frac{n I ( x - y )}{∣ x - y ∣}

= \frac{n I ( x - y )}{∣ x - y ∣}

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What is (\partial)/(\partial x)(In|x-y|) ?
The answer to (\partial)/(\partial x)(In|x-y|) is (nI(x-y))/(|x-y|)