What is the derivative of e^tsin(5t) ?

The derivative of e^tsin(5t) is e^tsin(5t)+cos(5t)*5e^t

What is the first derivative of e^tsin(5t) ?

The first derivative of e^tsin(5t) is e^tsin(5t)+cos(5t)*5e^t

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derivative of

e^{t} sin (5 t)

e^{t} sin (5 t) + cos (5 t) \cdot 5 e^{t}

Solution steps

\frac{d}{d t} (e^{t} sin (5 t))

Apply the Product Rule:

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

f = e^{t}, g = sin (5 t)

= \frac{d}{d t} (e^{t}) sin (5 t) + \frac{d}{d t} (sin (5 t)) e^{t}

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

\frac{d}{d t} (sin (5 t)) = cos (5 t) \cdot 5

= e^{t} sin (5 t) + cos (5 t) \cdot 5 e^{t}

What is the derivative of e^tsin(5t) ?
The derivative of e^tsin(5t) is e^tsin(5t)+cos(5t)*5e^t
What is the first derivative of e^tsin(5t) ?
The first derivative of e^tsin(5t) is e^tsin(5t)+cos(5t)*5e^t