Explanation:

The given equation is:

sin(xy) = log(xy) dy/dx

To solve this, we need to use implicit differentiation.

Steps:

Step 1: Take the derivative of both sides with respect to x using the chain rule.

d/dx(sin(xy)) = d/dx(log(xy) dy/dx)

Step 2: Simplify the left side using the chain rule and product rule.

cos(xy) * y + x * cos(xy) * y' = (1/xy) * (y' * x + y)

Step 3: Simplify the right side by distributing the derivative.

cos(xy) * y + x * cos(xy) * y' = (1/xy) * x * y' + (1/xy) * y

Step 4: Combine like terms on both sides.

cos(xy) * y - (1/xy) * y = (1/xy) * x * y' - x * cos(xy) * y'

Step 5: Factor out y' on the right side.

cos(xy) * y - (1/xy) * y = y' * (1/xy * x - x * cos(xy))

Step 6: Divide both sides by (1/xy * x - x * cos(xy)).

y' = [cos(xy) * y - (1/xy) * y] / [1/xy * x - x * cos(xy)]

Therefore, the solution is:

y' = [cos(xy) * y - (1/xy) * y] / [1/xy * x - x * cos(xy)]

Conclusion:

The derivative of sin(xy) is found to be [cos(xy) * y - (1/xy) * y] / [1/xy * x - x * cos(xy)].