What is (\partial)/(\partial x)(2sqrt(x*y)-1/2 x^2) ?

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

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(\partial)/(\partial x)(2sqrt(x*y)-1/2 x^2)

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

\frac{y}{x} - x

Solution steps

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

Simplify

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

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

= \frac{y}{x} - x

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