What is the d/(dt)(e^{-2t}{y}(t)) ?

The d/(dt)(e^{-2t}{y}(t)) is -2e^{-2t}{y}(t)+e^{-2t} d/(dt)({y}(t))

What is the first d/(dt)(e^{-2t}{y}(t)) ?

The first d/(dt)(e^{-2t}{y}(t)) is -2e^{-2t}{y}(t)+e^{-2t} d/(dt)({y}(t))

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\frac{d}{d t} (e^{- 2 t} y (t))

- 2 e^{- 2 t} y (t) + e^{- 2 t} \frac{d}{d t} (y (t))

Solution steps

\frac{d}{d t} (e^{- 2 t} y (t))

Apply the Product Rule:

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

f = e^{- 2 t}, g = y (t)

= \frac{d}{d t} (e^{- 2 t}) y (t) + \frac{d}{d t} (y (t)) e^{- 2 t}

\frac{d}{d t} (e^{- 2 t}) = - 2 e^{- 2 t}

= (- 2 e^{- 2 t}) y (t) + \frac{d}{d t} (y (t)) e^{- 2 t}

Simplify

= - 2 e^{- 2 t} y (t) + e^{- 2 t} \frac{d}{d t} (y (t))

What is the d/(dt)(e^{-2t}{y}(t)) ?
The d/(dt)(e^{-2t}{y}(t)) is -2e^{-2t}{y}(t)+e^{-2t} d/(dt)({y}(t))
What is the first d/(dt)(e^{-2t}{y}(t)) ?
The first d/(dt)(e^{-2t}{y}(t)) is -2e^{-2t}{y}(t)+e^{-2t} d/(dt)({y}(t))