b) Show that \(\displaystyle \frac{d}{dt}e^{rt}=re^{rt}\).

For (a) I am stuck because \(\displaystyle e^{(r_{1}+r_{2})t}=e^{(\lambda+i\mu+\lambda-i\mu)t}=e^{2\lambda\\t}\). I am not sure where to go from here. Should I just consider to arbitrary positive complex number rather than conjugate roots? I am considering roots because that is all that is talked about in the section.

For (b) do I just differentiate \(\displaystyle e^{\lambda\\t}(cos(\mu\\t)+isin(\mu\\t)\) since \(\displaystyle e^{(\lambda+i\mu)t}=e^{\lambda\\t}(cos(\mu\\t)+isin(\mu\\t)\) or can I just look at the power series expansion of \(\displaystyle e^{(\lambda+i\mu)t}\) and differentiate that?