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

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

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(\partial)/(\partial y)(xysqrt(x^2+y^2))

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\int x 3 x^{2} + 4 d x

\int \frac{5 x}{x ^{2} + 1} d x

n = 1 \sum \infty \frac{1}{n 3 ^{n}}

{\frac{1}{s ^{3}}}

\int sin^{3} (15 x) d x

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