You can turn any function with N parameters into N non-additive derivatives:
* Take the regular additive derivative of the function N times, this gives you N equations
* Solve for each of the parameters and your input x across the equations
* The parameters are the derivatives

eg: a x^2 = y
Take the derivative twice
2 a = y''
So we have
a = y'' / 2

We can also solve this by first solving the original equation to get x:
x = +- (y / a)^(1/2)
Then plug that in and solve to get the derivative (squaring removes the +-):
a = y'^2 / (4 y)

Since we have a 'y' in the denominator in that last parameteric derivative, we have singularities, and that's unattractive. Instead lets do this:
eg: a x^2 + b x = y
Take the derivative twice:
2 a = y''
So
a = y'' / 2
Then we can solve the original equation: x = (-b +- sqrt(b^2 + 4 a y) / (2 a))
And take the derivative of the original equation once:
2 a x + b = y
Plug stuff in and solve for b:
b = +- sqrt(y'^2 - 2 y y'')

eg: For the regular additive derivative its really simple:
a x = y
Take the additive derivative:
a = y'
Which is already solved

eg: a^x = y
Solve for x:
x = ln y / ln a
Take the derivative, plug in for x, and solve for a:
a = exp(y' / y)
Which is the regular multiplicative derivative

eg: x^a = y
Solve for x:
x = y^(1/a)
Take the derivative, plug in for x, and solve for a:
a = ln y / W(y ln(y) / y')
This gives you the exponential

eg: sin(a x) = y
Take the derivative twice:
y'' = -a^2 sin(a x) = -a^2 y
Solve for a:
a = sqrt(- y''/y)
Which gives you the equivalent oscillation rate for a 0-phase sine wave best matching your function y