What is the derivative of e^{-x}sin(2x) ?

The derivative of e^{-x}sin(2x) is -e^{-x}sin(2x)+2e^{-x}cos(2x)

What is the first derivative of e^{-x}sin(2x) ?

The first derivative of e^{-x}sin(2x) is -e^{-x}sin(2x)+2e^{-x}cos(2x)

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\frac{d}{d x} (e^{- x} sin (2 x))

- e^{- x} sin (2 x) + 2 e^{- x} cos (2 x)

Solution steps

\frac{d}{d x} (e^{- x} sin (2 x))

Apply the Product Rule:

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

f = e^{- x}, g = sin (2 x)

= \frac{d}{d x} (e^{- x}) sin (2 x) + \frac{d}{d x} (sin (2 x)) e^{- x}

\frac{d}{d x} (e^{- x}) = - e^{- x}

\frac{d}{d x} (sin (2 x)) = cos (2 x) \cdot 2

= (- e^{- x}) sin (2 x) + cos (2 x) \cdot 2 e^{- x}

Simplify

= - e^{- x} sin (2 x) + 2 e^{- x} cos (2 x)

What is the derivative of e^{-x}sin(2x) ?
The derivative of e^{-x}sin(2x) is -e^{-x}sin(2x)+2e^{-x}cos(2x)
What is the first derivative of e^{-x}sin(2x) ?
The first derivative of e^{-x}sin(2x) is -e^{-x}sin(2x)+2e^{-x}cos(2x)