What is (\partial)/(\partial x)(|xy|^{1/2}) ?

The answer to (\partial)/(\partial x)(|xy|^{1/2}) is (x|y|^{1/2})/(2|x|^{3/2)}

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(\partial)/(\partial x)(|xy|^{1/2})

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

\frac{x ∣ y ∣ ^{\frac{1}{2}}}{2 ∣ x ∣ ^{\frac{3}{2}}}

Solution steps

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

Simplify

∣ x y ∣^{\frac{1}{2}} : (∣ x ∣ ∣ y ∣)^{\frac{1}{2}}

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{2 ( ∣ x ∣ ∣ y ∣ ) ^{\frac{1}{2}}} \cdot \frac{x ∣ y ∣}{∣ x ∣} : \frac{x ∣ y ∣ ^{\frac{1}{2}}}{2 ∣ x ∣ ^{\frac{3}{2}}}

= \frac{x ∣ y ∣ ^{\frac{1}{2}}}{2 ∣ x ∣ ^{\frac{3}{2}}}

What is (\partial)/(\partial x)(|xy|^{1/2}) ?
The answer to (\partial)/(\partial x)(|xy|^{1/2}) is (x|y|^{1/2})/(2|x|^{3/2)}