Second Partial Derivative wrt r of ln (r^2 + s)
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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 |
$\blacksquare$
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: Exercise $2$