Why does $[f'(a)(a+\Delta x-a)+b]-[f'(a)(a-a)+b]=f'(a)\Delta x$ Why does $[f'(a)(a+\Delta x-a)+b]-[f'(a)(a-a)+b]=f'(a)\Delta x$? If I try to calculate this I get $f'(a)(\Delta x-1) \neq f'(a)\Delta x$.

Some of the terms cancel. Look:
$$\begin{align}
[f'(a)(\color{red}{a}+\Delta x\color{red}{-a})+b]-[f'(a)\color{blue}{(a-a)}+b]&= [f'(a)(\Delta x)+b]-[f'(a)\cdot \color{blue}{0}+b] \\
&= f'(a)(\Delta x) \color{green}{+b-b} \\
&= f'(a)(\Delta x).\end{align} $$
