A non-dimensionalization problem

[jimmy1066]

Homework Statement

A function $$f(x)$$ has dimensions of length, $$L$$.
I need to non-dimensionalize $$f(x)$$ using the substitution $$x=zL$$ and then differentiate it with respect to $$z$$.

Homework Equations

$$x=zL$$

The Attempt at a Solution

First I made the substitution:
$$f(x) \Rightarrow f(zL)$$

$$\frac{\partial f(zL)}{\partial z}$$ = L f '(z L)

Is this correct? I'm not sure because this now has dimensions of length.
What happens if I write: f(x) = f(zL) = f(x(z)) = F(z) and then differentiate w.r.t. z?

 

[hunt_mat]

I would go about it in the following way: define f(x)=Lg(x), then using the substitution for x as x=Lz, then:

$$f(x)=Lg(Lz)$$
So

$$f'(z)=Lg'(Lz)(Lz)'=L^{2}g'(Lz)$$