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

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

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

\frac{\partial}{\partial x} (x 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}}

\frac{\partial}{\partial x} (4 x (5 + y)^{- 1})

y^{^{'}} + 2 x y = x^{3}

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

\frac{d}{d x} (x (x^{2} - 4))

f (x) = x^{3} + 2 x, at x = 0

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