find all second - order partial derivatives for the function ( r=ln|2x + 3y| ).\n( r_{xx}=-\frac{4}{(2x +…

find all second - order partial derivatives for the function ( r=ln|2x + 3y| ).\n( r_{xx}=-\frac{4}{(2x + 3y)^{2}} )\n( r_{xy}=)

find all second - order partial derivatives for the function ( r=ln|2x + 3y| ).\n( r_{xx}=-\frac{4}{(2x + 3y)^{2}} )\n( r_{xy}=)

Answer

Explanation:

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

Using 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_{xy})

Differentiate (r_x=\frac{2}{2x + 3y}) with respect to (y). Using the quotient rule ((\frac{f}{g})^\prime=\frac{f^\prime g - fg^\prime}{g^{2}}), here (f = 2), (f^\prime=0), (g=2x + 3y), (g^\prime = 3). So (r_{xy}=\frac{0\times(2x + 3y)-2\times3}{(2x + 3y)^{2}}=-\frac{6}{(2x + 3y)^{2}}).

Answer:

(r_{xy}=-\frac{6}{(2x + 3y)^{2}})