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.