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

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

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

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

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

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

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

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

Simplify

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

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

a re a f (X) = 2 X - 8, 7, 10

\frac{d}{d x} (- 8 sin (2 x))

d er i v a t i v e g (x) = 2 - x

d er i v a t i v e f (x) = y^{\frac{2}{3}}

\int \frac{x ^{2} + 8 x + 4}{x ^{4} + 2 x ^{3} + x ^{2}} d x

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