What is the derivative of e^{-3x}sin(5x) ?

The derivative of e^{-3x}sin(5x) is -3e^{-3x}sin(5x)+5e^{-3x}cos(5x)

What is the first derivative of e^{-3x}sin(5x) ?

The first derivative of e^{-3x}sin(5x) is -3e^{-3x}sin(5x)+5e^{-3x}cos(5x)

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derivative of

e^{- 3 x} sin (5 x)

- 3 e^{- 3 x} sin (5 x) + 5 e^{- 3 x} cos (5 x)

Solution steps

\frac{d}{d x} (e^{- 3 x} sin (5 x))

Apply the Product Rule:

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

f = e^{- 3 x}, g = sin (5 x)

= \frac{d}{d x} (e^{- 3 x}) sin (5 x) + \frac{d}{d x} (sin (5 x)) e^{- 3 x}

\frac{d}{d x} (e^{- 3 x}) = - 3 e^{- 3 x}

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

= (- 3 e^{- 3 x}) sin (5 x) + cos (5 x) \cdot 5 e^{- 3 x}

Simplify

= - 3 e^{- 3 x} sin (5 x) + 5 e^{- 3 x} cos (5 x)

What is the derivative of e^{-3x}sin(5x) ?
The derivative of e^{-3x}sin(5x) is -3e^{-3x}sin(5x)+5e^{-3x}cos(5x)
What is the first derivative of e^{-3x}sin(5x) ?
The first derivative of e^{-3x}sin(5x) is -3e^{-3x}sin(5x)+5e^{-3x}cos(5x)