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:

<br /> f(x)=Lg(Lz)<br />
So

<br /> f&#039;(z)=Lg&#039;(Lz)(Lz)&#039;=L^{2}g&#039;(Lz)<br />