What is (\partial)/(\partial y)(1/(sqrt(xy))) ?

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

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(\partial)/(\partial y)(1/(sqrt(xy)))

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

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

Solution steps

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

Treat

x

as a constant

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{1}{2 ( x y ) ^{\frac{3}{2}}} \frac{\partial}{\partial y} (x y)

= - \frac{1}{2 ( x y ) ^{\frac{3}{2}}} \frac{\partial}{\partial y} (x y)

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

= - \frac{1}{2 ( x y ) ^{\frac{3}{2}}} x

Simplify

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

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

\int \frac{x}{1 + x ^{2}} d x

\int \frac{3}{x ^{2}} - \frac{2}{x ^{3}} d x

x^{3}

\frac{d}{d x} (ln (x) e^{x})

\frac{d}{d x} ((5 x - 4)^{2})

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