What is (\partial)/(\partial y)(sqrt(x^2+6xy+y^4)) ?

The answer to (\partial)/(\partial y)(sqrt(x^2+6xy+y^4)) is (2y^3+3x)/(sqrt(x^2+6xy+y^4))

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(\partial)/(\partial y)(sqrt(x^2+6xy+y^4))

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

\frac{2 y ^{3} + 3 x}{x ^{2} + 6 x y + y ^{4}}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

\frac{\partial}{\partial y} (x^{2} + 6 x y + y^{4}) = 4 y^{3} + 6 x

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

Simplify

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

= \frac{2 y ^{3} + 3 x}{x ^{2} + 6 x y + y ^{4}}

What is (\partial)/(\partial y)(sqrt(x^2+6xy+y^4)) ?
The answer to (\partial)/(\partial y)(sqrt(x^2+6xy+y^4)) is (2y^3+3x)/(sqrt(x^2+6xy+y^4))