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

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

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

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

\frac{x ^{\frac{1}{2}} + y ^{\frac{1}{2}}}{x ^{\frac{1}{2}}}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (x^{0.5} + y^{0.5}) = \frac{0.5}{x ^{\frac{1}{2}}}

= 2 (x^{0.5} + y^{0.5}) \frac{0.5}{x ^{\frac{1}{2}}}

Simplify

2 (x^{\frac{1}{2}} + y^{\frac{1}{2}}) \frac{0.5}{x ^{\frac{1}{2}}} : \frac{x ^{\frac{1}{2}} + y ^{\frac{1}{2}}}{x ^{\frac{1}{2}}}

= \frac{x ^{\frac{1}{2}} + y ^{\frac{1}{2}}}{x ^{\frac{1}{2}}}

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