What is (\partial)/(\partial x)(sqrt(xy^2+y^2)) ?

The answer to (\partial)/(\partial x)(sqrt(xy^2+y^2)) is y/(2sqrt(x+1))

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

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

\frac{y}{2 x + 1}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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