As the other answers have said, there's probably a typo in the problem statement, since
$$\frac{\mathrm d}{\mathrm dx}(6x^{100}-x^{55}+x)=600x^{99}−55x^{54}+1,$$
rather than the other way around. If you got that
$$\frac{\mathrm d}{\mathrm dx}(600x^{99}−55x^{54}+1)=59400x^{98}-2970x^{53},$$
then you differentiated correctly. As for your other questions:
1) In general, when you get stuck on any sort of math problem it helps to divide things into manageable chunks. When it comes to differentiating a function, look for smaller functions inside that you already know how to differentiate, such as trig functions, logs, exponential functions, and things of the form $x^n$. Then use the rules you learned in calculus to combine them:
$$\begin{align}
(x^n)'&=nx^{n-1}\\
\big(k\cdot f(x)\big)'&=k\cdot f'(x)\\
\big(f(x)+g(x)\big)'&=f'(x)+g'(x)\\
\big(f(x)\cdot g(x)\big)'&=f'(x)\cdot g(x)+f(x)\cdot g'(x)\\
\big(f(g(x))\big)'&=f'(g(x))\cdot g'(x),
\end{align}$$
where $f$ and $g$ are functions, and $k$ and $n$ are constants.
2) As for this case specifically, there are three terms being added and you (should) know how to differentiate each individually. Thus there's no need to get fancy and use the product rule or chain rule. You simply use the power rule a few times, and add up what you get.