partial derivative of ln(x+sqrt(x^2+y^2))

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∂∂x (ln(x+√x²+y²))=1√x²+y²

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∂ ∂ x (ln(x+√x²+y²))=1√x²+y²

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∂ ∂ x (ln(x+√x²+y²))

Treat y as a constant

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Apply the chain rule: 1x+√x²+y² ∂ ∂ x (x+√x²+y²)
=1x+√x²+y² ∂ ∂ x (x+√x²+y²)

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∂ ∂ x (x+√x²+y²)=1+x√x²+y²
=1x+√x²+y² (1+x√x²+y² )

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Simplify 1x+√x²+y² (1+x√x²+y² ): 1√x²+y²
=1√x²+y²