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

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

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(\partial)/(\partial x)(x*ln(x^2+y^2))

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

x^{3} \frac{d y}{d x} - x^{2} y - y^{3} = 0

\frac{d}{d x} (\frac{5}{1 - 9 x})

cos (x^{2} - 5)

\frac{d ^{2}}{d x ^{2}} (x^{3} y^{2} - 2 x^{2} y + 3 x)

\frac{x}{4}

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