# 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)$$\displaystyle{F}{\left({t}\right)}={e}^{{{\left({1}−{i}\right)}{t}}}$$
(b)$$\displaystyle{F}{\left({t}\right)}={e}^{{{3}{i}{t}}}$$
(c) $$\displaystyle{F}{\left({t}\right)}={e}^{{{\left({2}+{3}{i}\right)}{t}}}$$

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comentezq

(a)Let $$\displaystyle{F}{\left({t}\right)}={e}^{{{\left({1}-{i}\right)}{t}}}$$. Then $$F'(t)=(1-i)e^{(1-i)t}$$ and $$\displaystyle{F}'{\left({t}\right)}={\left({1}-{i}\right)}^{{{2}}}{e}^{{{\left({1}-{i}\right)}{t}}}=-{2}{i}{e}^{{{\left({1}-{i}\right)}{t}}}$$.
(b) Let $$\displaystyle{F}{\left({t}\right)}={e}^{{{3}{i}{t}}}$$. Then $$\displaystyle{F}'{\left({t}\right)}={3}{i}{e}^{{{3}{i}{t}}}$$ and $$\displaystyle{F}{''}{\left({t}\right)}={\left({3}{i}\right)}^{{{2}}}{e}^{{{3}{i}{t}}}=-{9}{e}^{{{3}{i}{t}}}$$.
(c) Let $$\displaystyle{F}{\left({t}\right)}={e}^{{{\left({2}+{3}{I}\right)}{t}}}$$. Then $$\displaystyle{F}'{\left({t}\right)}={\left({2}+{3}{i}\right)}{e}^{{{\left({2}+{3}{i}\right)}{t}}}$$ and $$\displaystyle{F}{''}{\left({t}\right)}={\left({2}+{3}{i}\right)}^{{{2}}}{e}^{{{\left({2}+{3}{i}\right)}{t}}}={\left(-{5}+{12}{i}\right)}{e}^{{{\left({2}+{3}{i}\right)}{t}}}$$.