Solution:

To find dy/dx, we need to differentiate the given equation with respect to x. Let's break down the solution into steps:

Step 1: Write down the given equation:
x^3 * x^2 * y * x * y^2 * y^3 = 81

Step 2: Simplify the equation:
x^6 * y^6 * xy^2 = 81

Step 3: Take the natural logarithm of both sides of the equation:
ln(x^6 * y^6 * xy^2) = ln(81)

Step 4: Apply the logarithmic properties to simplify the equation:
ln(x^6) + ln(y^6) + ln(xy^2) = ln(81)

Step 5: Use the logarithmic properties to simplify further:
6ln(x) + 6ln(y) + ln(x) + 2ln(y) = ln(81)

Step 6: Combine like terms:
7ln(x) + 8ln(y) = ln(81)

Step 7: Differentiate both sides of the equation with respect to x:
(7ln(x))' + (8ln(y))' = (ln(81))'

Step 8: Apply the chain rule to differentiate the natural logarithm terms:
(7/x) + (8/y) * (dy/dx) = 0

Step 9: Rearrange the equation to solve for dy/dx:
(8/y) * (dy/dx) = -(7/x)

Step 10: Multiply both sides by y/8:
dy/dx = -(7/8) * (y/x)

Therefore, the derivative of y with respect to x, dy/dx, is equal to -(7/8) * (y/x).