Let's calculate
The tools we've got are the relations :



Now we can make it :



Let's introduce another function :



Now we can find :

I think your proof contains a few mistakes. Here is my version.
It is important to understand that:

means and that means

We are asked to use the relations :

--------1

-------------------2

--------------3

Using the Leibnitz rule (1) gives:

--------------4

At this point we need to find an expression for

Let

Using the relation 2 above

-----------------5

using relation 3 above



So in 4



Substituting into 4 gives