real analysis - Derivative of a piecewise defined function

I want to ask something about the definition of derivative. If we have a function like this $f(x) = \begin{cases} x^2+1 & \quad \text{if } x<0 \\ \cos x & \quad \text{if } x\ge 0\\ \end{cases}$
and we want to compute the derivetive in $x_0=0$ why is necessary to compute this with the definition of derivative? It will be wrong to find the the derivative of $x^2+1$ and of $\cos x$ with types and to compute in $x_0=0$? Could you give me an example which is compute only with the limit and it is not compute with formulas? Thanks in advance