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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

\int \frac{1}{x ^{3} + x y ^{2}} d y

\frac{d}{d x} (- 16 sin (4 x))

\frac{d}{d x} (ln (\frac{1 + x}{x ^{3}}))

\int_{\frac{1}{5}}^{7} 8 x ln (5 x) d x

y^{^{'}} + \frac{2}{x} y = 4 x

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