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

The answer to (\partial)/(\partial y)(xyln(x^2+y^2)) is x(ln(x^2+y^2)+(2y^2)/(x^2+y^2))

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

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

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

Solution steps

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

Apply the Product Rule:

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

f = y, g = ln (x^{2} + y^{2})

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

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

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

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

Simplify

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

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

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