What is the d/(dt)(e^t(cos(t)-sin(t))) ?

The d/(dt)(e^t(cos(t)-sin(t))) is -2e^tsin(t)

What is the first d/(dt)(e^t(cos(t)-sin(t))) ?

The first d/(dt)(e^t(cos(t)-sin(t))) is -2e^tsin(t)

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

- 2 e^{t} sin (t)

Solution steps

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{t}, g = cos (t) - sin (t)

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

\frac{d}{d t} (e^{t}) = e^{t}

\frac{d}{d t} (cos (t) - sin (t)) = - sin (t) - cos (t)

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

Simplify

e^{t} (cos (t) - sin (t)) + (- sin (t) - cos (t)) e^{t} : - 2 e^{t} sin (t)

= - 2 e^{t} sin (t)

What is the d/(dt)(e^t(cos(t)-sin(t))) ?
The d/(dt)(e^t(cos(t)-sin(t))) is -2e^tsin(t)
What is the first d/(dt)(e^t(cos(t)-sin(t))) ?
The first d/(dt)(e^t(cos(t)-sin(t))) is -2e^tsin(t)