Seemingly Simple Derivative (as a limit) Problem

luke8ball

I'm having trouble showing the following:

lim [f(ax)-f(bx)]/x = f'(0)(a-b)
x→0

I feel like this should be really easy, but am I missing something? I tried to use the definition of the derivative, but I know I can't just say f(ax)-f(bx) = (a-b)f(x).

Any ideas?

arildno

Try to add zero in your numerator in the shape f(0)-f(0), and see if you can rearrange it in a clever manner.

luke8ball

You mean so that I get:

[lim f(ax) - f(0)]/x - [lim f(bx) - f(0)]/x
x→0 x→0

I had thought about that, but I still don't see how that gives me af'(0) - bf'(0)...

arildno

Think chain rule..

arildno

A further hint:
Let g(x)=ax. Then, g(0)=0