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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

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

\int 3 x^{2} (x^{3} + 1)^{6} d x

\int x (x + 1)^{- 4} d x

\int (x^{8} + sec^{2} (x)) d x

f (x) = \frac{( x ^{3} + 9 )}{x ^{3}}

x \to 0 lim (tan (3 x))

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