What is the rate of change of y with respct to x at the point where x = 8 if x^3 - 1/y^2 = 3xy

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The rate of change of y with respect to x at the point where x = 8 has to be determined given that x^3 - 1/y^2 = 3xy.

Differentiate x^3 - 1/y^2 = 3xy implicitly.

`(d(x^3 - 1/y^2))/(dx) = (d(3xy))/(dx)`

=> `3x^2 + 1/y*(dy/dx) = 3y + 3x*(dy/dx)`

=> `(dy/dx)(1/y - 3x) = 3y - 3x^2`

=> `dy/dx = (3y - 3x^2)/(1/y - 3x)`

The rate of change is a function of x as well as y.

At x = 8, `512 - 1/y^2 = 24y` , this is a cubic equation with real root approximately -21.33. Substitute this value of y with the value of x in the equation of the derivative.

**The rate of change is approximately 10.64**

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