Answer: (3y² * dy/dx + 3x² * dy/dx - 9y) / 9.
Step-by-step explanation: To find dy/dx for the relation x³ + y³ = 9xy, we can use the chain rule. The chain rule states that if y is a function of x and x is a function of t, then the derivative of y with respect to t is equal to the derivative of y with respect to x times the derivative of x with respect to t.
In this case, we are given that y is a function of x and we want to find the derivative of y with respect to x. To do this, we can rewrite the given equation as follows:
x³ + y³ = 9xy
y³ = 9xy - x³
We can then take the derivative of both sides of the equation with respect to x to find dy/dx as follows:
3y² * dy/dx = 9y + 9x * dy/dx - 3x² * dy/dx
3y² * dy/dx = (9y + 9x * dy/dx) - 3x² * dy/dx
0 = (9y + 9x * dy/dx) - (3y² * dy/dx + 3x² * dy/dx)
0 = 9y + 9x * dy/dx - 3y² * dy/dx - 3x² * dy/dx
We can then solve this equation for dy/dx as follows:
0 = 9y + 9x * dy/dx - 3y² * dy/dx - 3x² * dy/dx
9x * dy/dx = 3y² * dy/dx + 3x² * dy/dx - 9y
9x * dy/dx = 3y² * dy/dx + 3x² * dy/dx - 9y
dy/dx = (3y² * dy/dx + 3x² * dy/dx - 9y) / 9x
dy/dx = (3y² * dy/dx + 3x² * dy/dx - 9y) / 9x
Therefore, the derivative of y with respect to x for the given equation is equal to (3y² * dy/dx + 3x² * dy/dx - 9y) / 9.
