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

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

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

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

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

Solution steps

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

Treat

x

as a constant

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

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

Simplify

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

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

\int \frac{9 x ^{2} + 5 x + 9}{( x ^{2} + 1 ) ^{2}} d x

ln (4 + x^{2})

\int_{1}^{2} 15 4 x^{2} - 3 d x

f (x) = x^{2} (1 - 6 x)

n = 0 \sum \infty 7 (- \frac{1}{7})^{n}

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