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}}}

n = 8 \sum \infty \frac{1}{4 ^{n}}

\int \frac{x ^{3}}{( x + 1 ) ^{3}} d x

\frac{d}{d x} (2 x (x + 4)^{3})

\frac{\partial}{\partial x} (x^{2} + 3 y^{2} - 6 y + 4 x y)

f (x) = - x^{2} + 2 x + 4, at x = 3

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