Question

find all second - order partial derivatives for the function ( r = ln|2x + 3y| ).( r_{xx}=)

Explanation:

Step1: Find the first - order partial derivative \(r_x\)

Use the chain rule. If \(r = \ln|u|\) where \(u=2x + 3y\), then \(\frac{\partial r}{\partial x}=\frac{1}{u}\cdot\frac{\partial u}{\partial x}\).
Since \(\frac{\partial u}{\partial x}=2\), we have \(r_x=\frac{2}{2x + 3y}\).

Step2: Find the second - order partial derivative \(r_{xx}\)

Differentiate \(r_x=\frac{2}{2x + 3y}=2(2x + 3y)^{-1}\) with respect to \(x\) again.
Using the power rule \((u^n)^\prime=nu^{n - 1}u^\prime\) (here \(n=-1\) and \(u = 2x+3y\), \(u^\prime = 2\)).
\(r_{xx}=2\times(-1)\times(2x + 3y)^{-2}\times2\).

Answer:

\(r_{xx}=-\frac{4}{(2x + 3y)^2}\)