altering Newton’s method

The usual scheme for Newton’s method is $z_{n+1} = z_{n} - \frac{f(z_{n})}{f'(z_{n})}$. But what if we just choose some random $g(z)$ for $\frac{f(z_{n})}{f'(z_{n})}$?

Set $g(z) = \frac{\cos(e^{z})}{e^{\sin(z)}}$
Now, as far as differential equations go, solving
$\frac{f(z)}{f'(z)} = \frac{\cos(e^{z})}{e^{\sin(z)}}$
is pretty tricky.

But I can still generate a fractal from it: