What is (\partial)/(\partial x)(4(y/x)^{1/2}) ?

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

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(\partial)/(\partial x)(4(y/x)^{1/2})

\frac{\partial}{\partial x} (4 (\frac{y}{x})^{\frac{1}{2}})

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

Solution steps

\frac{\partial}{\partial x} (4 (\frac{y}{x})^{\frac{1}{2}})

Treat

y

as a constant

Take the constant out:

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

= 4 \frac{\partial}{\partial x} ((\frac{y}{x})^{\frac{1}{2}})

Apply the chain rule:

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

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

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

= 4 \cdot \frac{1}{2 ( \frac{y}{x} ) ^{\frac{1}{2}}} (- \frac{y}{x ^{2}})

Simplify

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

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

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