Why is implicit differentiation necessary in order to find derivatives of curves which do not represent functions?

Come up with your own implicitly defined function. What is its first, second, and third derivative?

Implicit Differentiation is necessary in order to find derivatives of curves that do not represent a function because the functions cannot be simplified to a simple function. Using dy/dx and implicit differentiation can be used to find the derivative with respect to one variable and determine the affect each variable has on one another and calculate the rate of change of the variables over time. Differentiating both x and y in Implicit Differentiation makes it so that complex functions are much easier and instead of solving for one variable, differentiating one variable to the other is an easier method. My implicit function is 2y^2-x^2+x^3y=2. First Derivative = - 3yx^2+2x/4y+x^3 Second Derivative = 3yx^4-4y^3-24y^2x+8y / (4y+x^3) ^2 Third Derivative = 6y (-x^6+28yx^3+4y^2x^2-8x^2-16y) / (4y+x^3) ^3

The answer is implicit