What is (\partial)/(\partial x)(y/((2*sqrt(x*y)))) ?

The answer to (\partial)/(\partial x)(y/((2*sqrt(x*y)))) is -(y^{1/2})/(4x^{3/2)}

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(\partial)/(\partial x)(y/((2sqrt(xy))))

\frac{\partial}{\partial x} (\frac{y}{( 2 \cdot x \cdot y )})

- \frac{y ^{\frac{1}{2}}}{4 x ^{\frac{3}{2}}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

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

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

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

Simplify

\frac{y}{2} (- \frac{1}{2 ( x y ) ^{\frac{3}{2}}} y) : - \frac{y ^{\frac{1}{2}}}{4 x ^{\frac{3}{2}}}

= - \frac{y ^{\frac{1}{2}}}{4 x ^{\frac{3}{2}}}

y = e^{\frac{x}{3}}, at x = 0

\int cos^{2} (\frac{2 x}{3}) d x

n = 0 \sum \infty 2 + e^{- 3 n}

g (s) = \frac{5}{s - 3}

\frac{d}{d x} (- 2 x cos (x))

What is (\partial)/(\partial x)(y/((2*sqrt(x*y)))) ?
The answer to (\partial)/(\partial x)(y/((2*sqrt(x*y)))) is -(y^{1/2})/(4x^{3/2)}