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College
Calculus and Analysis
Differential Calculus
Derivatives

remolatg

Answered question

2021-06-23

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) $F (t) = e^{3 i t}$
(c) $F (t) = e^{(2 + 3 i) t}$

Answer & Explanation

comentezq

Skilled2021-06-24Added 106 answers

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

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