# Second Partial Derivative wrt r of ln (r^2 + s)

## Theorem

$\dfrac {\partial^2} {\partial r^2} \map \ln {r^2 + s} = \dfrac {2 \paren {s - r^2} } {\paren {r^2 + s}^2}$

## Proof

 $\ds \dfrac {\partial^2} {\partial r^2} \map \ln {r^2 + s}$ $=$ $\ds \map {\dfrac \partial {\partial r} } {\dfrac \partial {\partial r} \map \ln {r^2 + s} }$ Definition of Second Partial Derivative $\ds$ $=$ $\ds \dfrac \partial {\partial r} \frac 1 {r^2 + s} 2 r$ Derivative of Natural Logarithm, Chain Rule for Derivatives, treating $s$ as a constant $\ds$ $=$ $\ds \frac {2 \paren {r^2 + s} - 2 r \paren {2 r} } {\paren {r^2 + s}^2}$ Quotient Rule for Derivatives, again treating $s$ as a constant $\ds$ $=$ $\ds \frac {2 \paren {s - r^2} } {\paren {r^2 + s}^2}$ simplifying

$\blacksquare$
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