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

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

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(\partial)/(\partial x)(ln|xy|)

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

\frac{1}{x}

Solution steps

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

Simplify

ln ∣ x y ∣ : ln (∣ x ∣ ∣ y ∣)

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

= \frac{1}{∣ x ∣ ∣ y ∣} \cdot \frac{x ∣ y ∣}{∣ x ∣}

Simplify

\frac{1}{∣ x ∣ ∣ y ∣} \cdot \frac{x ∣ y ∣}{∣ x ∣} : \frac{1}{x}

= \frac{1}{x}

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