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

The answer to (\partial)/(\partial x)(sqrt(|xy|)) is (xsqrt(|y|))/(2|x|sqrt(|x|))

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

\frac{\partial}{\partial x} (∣ x y ∣)

\frac{x ∣ y ∣}{2 ∣ x ∣ ∣ x ∣}

Solution steps

\frac{\partial}{\partial x} (∣ x y ∣)

Treat

y

as a constant

Apply the chain rule:

\frac{1}{2 ∣ x y ∣} \frac{\partial}{\partial x} (∣ x y ∣)

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

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

= \frac{1}{2 ∣ x y ∣} \cdot \frac{x ∣ y ∣}{∣ x ∣}

Simplify

\frac{1}{2 ∣ x y ∣} \cdot \frac{x ∣ y ∣}{∣ x ∣} : \frac{x ∣ y ∣}{2 ∣ x ∣ ∣ x ∣}

= \frac{x ∣ y ∣}{2 ∣ x ∣ ∣ x ∣}

\frac{d y}{d x} + \frac{y}{x} = x^{2}

t^{2} + 9

x \to 2 lim (e^{x})

\int \frac{1}{6 - 36 x ^{2}} d x

\frac{2 - sec ( x )}{tan ( x )}

What is (\partial)/(\partial x)(sqrt(|xy|)) ?
The answer to (\partial)/(\partial x)(sqrt(|xy|)) is (xsqrt(|y|))/(2|x|sqrt(|x|))