# Find the derivatives F'(t) and F''(t) for each of the following complex-valued functions of the real variable t. (a) F(t)=e^{(1−i)t} (b) F(t)=e^{3 i t

Find the derivatives F'(t) and F''(t) for each of the following complex-valued functions of the real variable t.
(a)$F\left(t\right)={e}^{\left(1-i\right)t}$
(b)$F\left(t\right)={e}^{3it}$
(c) $F\left(t\right)={e}^{\left(2+3i\right)t}$

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(a)Let $F\left(t\right)={e}^{\left(1-i\right)t}$. Then ${F}^{\prime }\left(t\right)=\left(1-i\right){e}^{\left(1-i\right)t}$ and ${F}^{\prime }\left(t\right)={\left(1-i\right)}^{2}{e}^{\left(1-i\right)t}=-2i{e}^{\left(1-i\right)t}$.
(b) Let $F\left(t\right)={e}^{3it}$. Then ${F}^{\prime }\left(t\right)=3i{e}^{3it}$ and $F{}^{″}\left(t\right)={\left(3i\right)}^{2}{e}^{3it}=-9{e}^{3it}$.
(c) Let $F\left(t\right)={e}^{\left(2+3I\right)t}$. Then ${F}^{\prime }\left(t\right)=\left(2+3i\right){e}^{\left(2+3i\right)t}$ and $F{}^{″}\left(t\right)={\left(2+3i\right)}^{2}{e}^{\left(2+3i\right)t}=\left(-5+12i\right){e}^{\left(2+3i\right)t}$.