Proving difference quotient is equal to derivative at 0 everyone, thankfully I was able to figure out parts 1 and parts 3.  However, I am stuck on part 2. The conditions somewhat confuse me.  All I so far was (first steps are similar to part 1):
Start with: 
$f(a_n)=f(0)+a_n(f'(0)+c_n  $
$f(b_n)=f(0)+b_n(f'(0))+d_n   $
such that $|c_n/a_n| $ goes to 0 and $|dn/bn| $ goes to 0.
From there I get plugging in for $f(b_n)$ and $f(a_n)$: 
$D_n=f'(0)+(d_n-c_n)/(b_n-a_n)  $
So now I somehow need to show that:
$\lim (d_n-c_n)/(b_n-a_n)=0     $ to show that $D_n=f'(0)$, using the conditions $0<a_n<b_n$ and $b_n/(b_n-a_n)$.  The problem is I'm not sure how to use these conditions to show that the limit goes to 0.  For part 1, it was pretty simple using Squeeze Theorem.  However, I can't figure out how to do it now with these new conditions.  Any help would be much appreciated.  Thank you.  

Note that $$0\leq\left|\frac{d_n-c_n} {b_n-a_n} \right|\leq\frac{|d_n|+|c_n|} {b_n-a_n} =\left(\frac{|d_n|} {b_n} +\frac{|c_n|} {b_n} \right) \frac{b_n} {b_n-a_n} \leq \left(\frac{|d_n|} {b_n} +\frac{|c_n|} {a_n} \right) M$$ and now the last expression tends to $(0+0)M=0$ so that by squeeze theorem $|d_{n} - c_n|/(b_n-a_n) \to 0$ and your job is done. 
