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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

= 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}

= 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}}

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

Simplify

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

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

\int 3 x sin (5 x) d x

x = 9 y^{2}, x = 28 + 2 y^{2}

\int - e^{t} d t

f (x) = 3 - 6 x^{2}, - 57, at x = 3

x \to 3 - lim (ln (x - 3))

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