find the second order derivative, rxy(x,y) for r(x,y)=xy/8x+5y
To find the second-order derivative of the function r(x, y), which is denoted as rxy(x, y), we'll need to differentiate the function twice with respect to x and then y. Here's how you can do it step by step:
Step 1:
Start by finding the first-order derivative of r(x, y) with respect to x. To do this, treat y as a constant and differentiate xy/8x+5y with respect to x. The derivative of xy with respect to x is y, and the derivative of 8x+5y with respect to x is 8. Thus, the first derivative with respect to x is:
r'x(x, y) = y/8
Step 2:
Next, differentiate the first-order derivative obtained in step 1, r'x(x, y), with respect to y. Treat x as a constant this time, and differentiate y/8 with respect to y. The derivative of y with respect to y is 1 and the derivative of any constant (in this case, 8) with respect to y is 0. Thus, the second derivative with respect to y is:
r'xy(x, y) = 1/8
So the second-order derivative of r(x, y) with respect to x and then y, or rxy(x, y), is 1/8.
In summary:
rxy(x, y) = 1/8