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

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

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

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

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

Solution steps

\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) = y

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

Simplify

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

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

\int_{2}^{4} 3 x - 6 d x

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

\frac{\partial}{\partial x} (5 x^{2} + x y + 2 y^{2})

x \to 4 - lim (\frac{4}{x - 4})

\frac{d}{d y} (\frac{y ^{3}}{39} + \frac{13}{4 y})

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