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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (400 (15 - x^{2}) (3 x - 2))

\int cos (b x) d x

\int 2 x^{3} + sin (x) d x

\int (x + 4) (x - 6) d x

\frac{d}{d x} (\frac{x ^{2}}{arccos ( x ^{2} )})

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