Let $$f(x) = \begin{cases} 5x-3x^{2} & \text{for } x < 0\;, \\ 4x^{2}+5x & \text{for } x \ge 0\;. \end{cases}$$ According to the definition of the derivative, to compute $f'(0)$, we need to compute the left-hand limit
$\lim\limits_{x\to 0^-}$ , which is ,
and the right-hand limit
$\lim\limits_{x\to 0^+}$ , which is .
We conclude that $f'(0)$ is

Note:  If a limit or derivative is undefined, enter 'undefined' as your answer.