What is (\partial)/(\partial t)(e^{-t}sin(t)) ?

The answer to (\partial)/(\partial t)(e^{-t}sin(t)) is -e^{-t}sin(t)+e^{-t}cos(t)

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(\partial)/(\partial t)(e^{-t}sin(t))

\frac{\partial}{\partial t} (e^{- t} sin (t))

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

Solution steps

\frac{\partial}{\partial t} (e^{- t} sin (t))

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial t} (e^{- t}) = - e^{- t}

\frac{\partial}{\partial t} (sin (t)) = cos (t)

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

Simplify

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

What is (\partial)/(\partial t)(e^{-t}sin(t)) ?
The answer to (\partial)/(\partial t)(e^{-t}sin(t)) is -e^{-t}sin(t)+e^{-t}cos(t)