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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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\frac{1}{1 + sin ( x )}

\frac{1}{( 1 - 0.8 \cdot cos ^{2} ( x ) )}

\frac{5}{( s ^{2} + 5 )}

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