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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = x^{2} + y^{2}

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

\frac{\partial}{\partial x} (x) = 1

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

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

Simplify

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

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

\frac{\partial}{\partial x} (7 - x^{3} - y^{7})

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\int (4 x - csc^{2} (x)) d x

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