What is (\partial)/(\partial y)(sqrt(3x^2+8xy+y^2)) ?

The answer to (\partial)/(\partial y)(sqrt(3x^2+8xy+y^2)) is (y+4x)/(sqrt(3x^2+8xy+y^2))

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(\partial)/(\partial y)(sqrt(3x^2+8xy+y^2))

\frac{\partial}{\partial y} (3 x^{2} + 8 x y + y^{2})

\frac{y + 4 x}{3 x ^{2} + 8 x y + y ^{2}}

Solution steps

\frac{\partial}{\partial y} (3 x^{2} + 8 x y + y^{2})

Treat

x

as a constant

Apply the chain rule:

\frac{1}{2 3 x ^{2} + 8 x y + y ^{2}} \frac{\partial}{\partial y} (3 x^{2} + 8 x y + y^{2})

= \frac{1}{2 3 x ^{2} + 8 x y + y ^{2}} \frac{\partial}{\partial y} (3 x^{2} + 8 x y + y^{2})

\frac{\partial}{\partial y} (3 x^{2} + 8 x y + y^{2}) = 2 y + 8 x

= \frac{1}{2 3 x ^{2} + 8 x y + y ^{2}} (2 y + 8 x)

Simplify

\frac{1}{2 3 x ^{2} + 8 x y + y ^{2}} (2 y + 8 x) : \frac{y + 4 x}{3 x ^{2} + 8 x y + y ^{2}}

= \frac{y + 4 x}{3 x ^{2} + 8 x y + y ^{2}}

What is (\partial)/(\partial y)(sqrt(3x^2+8xy+y^2)) ?
The answer to (\partial)/(\partial y)(sqrt(3x^2+8xy+y^2)) is (y+4x)/(sqrt(3x^2+8xy+y^2))