What is the d/(dt)(e^{7t}e^{4it}) ?

The d/(dt)(e^{7t}e^{4it}) is (7e^{7t}cos(4t)-4e^{7t}sin(4t))+i(4e^{7t}cos(4t)+7e^{7t}sin(4t))

What is the first d/(dt)(e^{7t}e^{4it}) ?

The first d/(dt)(e^{7t}e^{4it}) is (7e^{7t}cos(4t)-4e^{7t}sin(4t))+i(4e^{7t}cos(4t)+7e^{7t}sin(4t))

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\frac{d}{d t} (e^{7 t} e^{4 i t})

(7 e^{7 t} cos (4 t) - 4 e^{7 t} sin (4 t)) + i (4 e^{7 t} cos (4 t) + 7 e^{7 t} sin (4 t))

Solution steps

\frac{d}{d t} (e^{7 t} e^{4 i t})

Simplify

e^{7 t} e^{4 i t} : e^{7 t} cos (4 t) + i e^{7 t} sin (4 t)

= \frac{d}{d t} (e^{7 t} cos (4 t) + i e^{7 t} sin (4 t))

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d t} (e^{7 t} cos (4 t)) + \frac{d}{d t} (i e^{7 t} sin (4 t))

\frac{d}{d t} (e^{7 t} cos (4 t)) = 7 e^{7 t} cos (4 t) - 4 e^{7 t} sin (4 t)

\frac{d}{d t} (i e^{7 t} sin (4 t)) = i (4 e^{7 t} cos (4 t) + 7 e^{7 t} sin (4 t))

= 7 e^{7 t} cos (4 t) - 4 e^{7 t} sin (4 t) + i (4 e^{7 t} cos (4 t) + 7 e^{7 t} sin (4 t))

Rewrite

7 e^{7 t} cos (4 t) - 4 e^{7 t} sin (4 t) + i (4 e^{7 t} cos (4 t) + 7 e^{7 t} sin (4 t))

in standard complex form:

(7 e^{7 t} cos (4 t) - 4 e^{7 t} sin (4 t)) + (4 e^{7 t} cos (4 t) + 7 e^{7 t} sin (4 t)) i

= (7 e^{7 t} cos (4 t) - 4 e^{7 t} sin (4 t)) + i (4 e^{7 t} cos (4 t) + 7 e^{7 t} sin (4 t))

What is the d/(dt)(e^{7t}e^{4it}) ?
The d/(dt)(e^{7t}e^{4it}) is (7e^{7t}cos(4t)-4e^{7t}sin(4t))+i(4e^{7t}cos(4t)+7e^{7t}sin(4t))
What is the first d/(dt)(e^{7t}e^{4it}) ?
The first d/(dt)(e^{7t}e^{4it}) is (7e^{7t}cos(4t)-4e^{7t}sin(4t))+i(4e^{7t}cos(4t)+7e^{7t}sin(4t))