I don't get a question that I have to answer.

"Determine if there is a derivative over the entire domain of the following functions. If not, explain why."

f(x)=
{4x^2+3, x < 0}
{6x+3, x > 0}

f(x)=
{3-x^3, x < 2}
{-4x^2+4x, x > 2}

I'll give you a hint in that you need to consider what happens when x = 0. That's the only potential value of discrepancy. Key questions: is the function continuous at x = 0? And if so, is it differentiable at x = 0 and what are the derivative values?

So I have to find the derivative of both the first and second equations (provided they are continuous?)?

Yes. Do that. That should answer your question for you.

So the answer to the first equation is "no derivative" because although the first piecewise is continuous, the slope of the first equation at 0 is 0, but the slope of the linear equation is 6. Is this right?

Would it be so that "no, there is no derivative over the entire domain because the shift in the derivative between the first and second equations is not the same."?