Is there a relationship between

[epkid08]

Let's say I setup the equation:

$$f(x) = f(x)$$

Now, let's say I add two independent real-valued variables, a and b, to the equation, where either a is a function of b and x or b is a function of a and x, making the statement true at all times:

$$f(x) = af(x+b)$$

Finding a' and b' we have:

$$a' = -\frac{f'(x+b)f(x)}{f(x+b)^2}$$

$$b' = -f^{-1}'(\frac{f(x)}{a})\frac{f(x)}{a}$$

My question is, is there a distinct relationship between a' and b'?

 

[epkid08]

I think I got it, they are reciprocals, but they don't readily cancel out.