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

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

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(\partial)/(\partial y)((xy)^{1/2})

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{1}{8 y ^{2}}

\frac{d}{d x} ((arctan (6 x))^{2})

x^{2} sin^{8} (x) + x cos^{- 3} (x)

f (x) = 2 x - 5

x \to 0 lim (\frac{5 x}{3 x + 7})

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