Detailed Video Solution: https://youtu.be/6aIuj2b2J38
This is one of my favorite questions, a similar kind of question was asked once in my IISc class by Prof Shevade. Surprisingly, very few people were able to provide an answer to it. (Sachin Mittal)
Simple and elegant Question.
Since $g$ is not differentiable we cannot use the product rule.
$$
f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}=\lim _{h \rightarrow 0} \frac{h g(h)}{h}=8 .
$$
Detailed Video Solution: https://youtu.be/6aIuj2b2J38