detailed working for differentiation

Hi, I've been working on my differentiation unit for the last couple of days and I am having a lot of trouble getting the correct answer.

as an example, for the question

of

i get an answer of

as apposed to the textbook answer of

I've been looking through my working to find the problem, and i've found the location where i go in a different direction from the proper answer, but i can't see how i would change anything to get the right answer. Since i can't find the right answer with my own working out, my question is could somebody please point out where i go wrong and correct it (preferably in detail.

here is my working out (numerator is unimportant so i am not showing the working for it)

the next 2 lines is where i suspect my mistake to be

lim h→0

which does not

Any help that is given would be greatly appreciated.