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

The answer to (\partial)/(\partial x)(ln((x^2+y^2)^2)) is (4x)/(x^2+y^2)

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

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

\frac{4 x}{x ^{2} + y ^{2}}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

\frac{1}{( x ^{2} + y ^{2} ) ^{2}} \frac{\partial}{\partial x} ((x^{2} + y^{2})^{2})

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

\frac{\partial}{\partial x} ((x^{2} + y^{2})^{2}) = 4 x (x^{2} + y^{2})

= \frac{1}{( x ^{2} + y ^{2} ) ^{2}} \cdot 4 x (x^{2} + y^{2})

Simplify

\frac{1}{( x ^{2} + y ^{2} ) ^{2}} \cdot 4 x (x^{2} + y^{2}) : \frac{4 x}{x ^{2} + y ^{2}}

= \frac{4 x}{x ^{2} + y ^{2}}

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What is (\partial)/(\partial x)(ln((x^2+y^2)^2)) ?
The answer to (\partial)/(\partial x)(ln((x^2+y^2)^2)) is (4x)/(x^2+y^2)