Root finding algorithm without using derivatives

This algorithm finds a root of an equation, like Newton's method, but is designed for situations when the function in question can be solved for x from a function f(x) though the derivative might be complicated or expensive to calculate making many other algorithms like Newton's method unsuitable...
Starting with an example function:

One first solves the equation for one x on the left from the original equation:

Then set y = 0 in this equation...

Then use this above formula and plug into a recursive structure like this:

Now you can plug in an initial guess (here -2):

And proceed recursively...

It approaches the root of 1.6666 quadratically...

**Proof**

Still working on a good proof...