What is (\partial)/(\partial x)(2(y/x)^{0.5}) ?

The answer to (\partial)/(\partial x)(2(y/x)^{0.5}) is -(y/(x^3))^{1/2}

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(\partial)/(\partial x)(2(y/x)^{0.5})

\frac{\partial}{\partial x} (2 (\frac{y}{x})^{0.5})

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

Solution steps

\frac{\partial}{\partial x} (2 (\frac{y}{x})^{0.5})

Treat

y

as a constant

Take the constant out:

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

= 2 \frac{\partial}{\partial x} ((\frac{y}{x})^{0.5})

Apply the chain rule:

\frac{0.5}{( \frac{y}{x} ) ^{0.5}} \frac{\partial}{\partial x} (\frac{y}{x})

= \frac{0.5}{( \frac{y}{x} ) ^{0.5}} \frac{\partial}{\partial x} (\frac{y}{x})

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

= 2 \cdot \frac{0.5}{( \frac{y}{x} ) ^{0.5}} (- \frac{y}{x ^{2}})

Simplify

2 \cdot \frac{0.5}{( \frac{y}{x} ) ^{\frac{1}{2}}} (- \frac{y}{x ^{2}}) : - (\frac{y}{x ^{3}})^{\frac{1}{2}}

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

- \frac{y}{( x ^{2} + y ^{2} )} d x + \frac{x}{x ^{2} + y ^{2}} d y = 0

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What is (\partial)/(\partial x)(2(y/x)^{0.5}) ?
The answer to (\partial)/(\partial x)(2(y/x)^{0.5}) is -(y/(x^3))^{1/2}