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})