Final answer:
The rate of change of y with respect to x when x=1 given that ln(xy)=x is 0, as calculated by differentiating both sides of the equation and simplifying.
Step-by-step explanation:
To find the rate of change of y with respect to x when x=1 given that ln(xy)=x, we need to differentiate both sides of the equation with respect to x. Applying the product rule to the left hand side, since y is a function of x, we get:
d/dx [ln(xy)] = d/dx [x]
1/y * (y + xy') = 1
Where y' denotes the derivative of y with respect to x. By simplifying, we solve for y':
y + xy' = y
xy' = 0
y' = 0 / x = 0
Thus, the rate of change of y with respect to x at x=1 is 0.