What is (\partial)/(\partial y)(sqrt(x^2+6xy+2y^2)) ?

The answer to (\partial)/(\partial y)(sqrt(x^2+6xy+2y^2)) is (2y+3x)/(sqrt(x^2+6xy+2y^2))

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(\partial)/(\partial y)(sqrt(x^2+6xy+2y^2))

\frac{\partial}{\partial y} (x^{2} + 6 x y + 2 y^{2})

\frac{2 y + 3 x}{x ^{2} + 6 x y + 2 y ^{2}}

Solution steps

\frac{\partial}{\partial y} (x^{2} + 6 x y + 2 y^{2})

Treat

x

as a constant

Apply the chain rule:

\frac{1}{2 x ^{2} + 6 x y + 2 y ^{2}} \frac{\partial}{\partial y} (x^{2} + 6 x y + 2 y^{2})

= \frac{1}{2 x ^{2} + 6 x y + 2 y ^{2}} \frac{\partial}{\partial y} (x^{2} + 6 x y + 2 y^{2})

\frac{\partial}{\partial y} (x^{2} + 6 x y + 2 y^{2}) = 4 y + 6 x

= \frac{1}{2 x ^{2} + 6 x y + 2 y ^{2}} (4 y + 6 x)

Simplify

\frac{1}{2 x ^{2} + 6 x y + 2 y ^{2}} (4 y + 6 x) : \frac{2 y + 3 x}{x ^{2} + 6 x y + 2 y ^{2}}

= \frac{2 y + 3 x}{x ^{2} + 6 x y + 2 y ^{2}}

What is (\partial)/(\partial y)(sqrt(x^2+6xy+2y^2)) ?
The answer to (\partial)/(\partial y)(sqrt(x^2+6xy+2y^2)) is (2y+3x)/(sqrt(x^2+6xy+2y^2))