How do I solve for lim h->0 (((a+h)^2 + 1)/(3(a+h)+7) - ((a^2 + 1)/3a + 7)) and then all divided by h?

Question

The question was use the limit definition to find the derivative of f(x)= (x^2 + 1)/(3x + 7).

I tried solving what I did above, but had no luck. I used wolfram alpha to get my answer of (3(a^2) + 14a - 3)/((3a + 7)^2). Could someone please show me the steps to arrive at this answer?

Answer 1

First of all, why did you switch from x to a ?

My first line ...
lim [ ((x+h)^2+1)/(3(x+h)+7) - (x^2 + 1)/(3x + 7) ] /h as h ---> 0
= lim [((3x+7)((x+h)^2 + 1) - (3x+3h+7)((x^2+1))/(((3x+3h+7)(3x+7))] / h

carefully expand the numerator, leave the bottom alone
Each term in the numerator WILL have a common factor of h, cancel it with the h at the bottom.
You will get the answer that your Wolfram link gave you.