The problem is this:

Find the derivative of the function

using a difference quotient. Hint: leave the denominator in factored form.

Also, I should mention, the only way my class has learned to solve for this so far is using the equation:

So, I believe I've solved the problem, but the way in which I did it seems incorrect, even though it yields the correct answer (or maybe there's a better way...I just feel like my way isn't what would be preferred!)

Here's what I did:

1.)

2.)

3.) I then found a common denominator and combined the top. I also made the h on the bottom into

to make it a little easier to write. That gave me this:

4.) For the sake of not writing out all the algebra, I multiplied the entire top part through and ended up with this:

5.) Then I factored out an "h" from the top and canceled it with the 1/h, getting this:

6.) Now here's where I think what I've done might not be considered correct. I got stuck once I reached this point, so I attempted to just take the limit of all of that at once like this (sorry for how long this is!):

Which, after figuring all of that out gave me this:

If you simplify that you get

Factor out a 4 from the top, cancel, and you get a final answer of

Now, that's the right answer but...am I getting to it correctly? That's my question.

Any help at all would be much appreciated! I know it's a really long problem.