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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (2 x z (x))

\frac{d}{d s} (\frac{s}{s ^{2} + 9})

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

\int 3 t e^{2 t} d t

x \to 1 lim (- 6 + 4 x^{2})

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