Calculating derivative by definition vs not by definition I'm not entirely sure I understand when I need to calculate a derivative using the definition and when I can do it normally. 
The following two examples confused me:
$$ g(x) = \begin{cases} x^2\cdot \sin(\frac {1}{x}) & x \neq 0 \\ 0 & x=0 \end{cases} $$
$$ f(x) = \begin{cases} e^{\frac {-1}{x}} & x > 0 \\ -x^2 & x\leq 0 \end{cases} $$
I understand that I can differentiate normally for any $x$ that's not 0 (in both of these examples). 
I'm confused because I saw an example with $f(x)$ in which they calculated the $f'(x)$ by differentiating:
$$ f'(x) = \begin{cases} \frac {1}{x^2}\cdot e^{\frac {-1}{x}}  & x > 0 \\ -2x & x\leq0 \end{cases} $$
and then they calculated $f'(0)$ not using the definition but by $ \lim_{x\to0^-} f'(x)$ $\lim_{x\to0^+} f'(x)$
For $g(x)$ though I know that $g'(0)$ exists (using definition)  but
$$g'(x) = \begin{cases} 2x\cdot \sin(\frac {1}{x})-\cos(\frac {1}{x})   & x \ne 0 \\ 0 & x=0 \end{cases}$$  
and you can't calculate $\lim_{x\to0^-} g'(x)$ or $\lim_{x\to0^+} g'(x)$ since $\lim_{x\to0^{+/-}} 2x\cdot \sin(\frac {1}{x})-\cos(\frac {1}{x}) $  doesn't exist. 
So what's the difference between these two? When can I just differentiate normally like in the first example ($f(x)$) and when do I have to use the definition like in the second example ($g(x)$)?
I'd appreciate the help. 
Edit: When I"m referring to the definition I'm referring to the following: 
$\lim_{h \rightarrow 0} \dfrac{f(x+h) - f(x)}{h}$

In what you quoted, the statement $f'(x)=-2x$ is valid only on $x<0$. The existence of $f'(0)$ must be determined from the definition. But read on.
Perhaps the following exercise will help you. Suppose $f$ is continuous on $(-a,a)$ and differentiable except perhaps at $0$. Suppose $$\lim_{x\to 0} f'(x)=L\,.$$
Then $f$ is differentiable at $0$ and $f'(0)=L$. (Hint: Use the Mean Value Theorem.)
Let me add the comment that the discussion in comments above begs the question: We can't say $f'$ is continuous at $0$ unless we first know it is defined there!
