What is (\partial)/(\partial x)(e^{-3x}sin(2t-3x)) ?

The answer to (\partial)/(\partial x)(e^{-3x}sin(2t-3x)) is -3e^{-3x}sin(2t-3x)-3e^{-3x}cos(2t-3x)

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(\partial)/(\partial x)(e^{-3x}sin(2t-3x))

\frac{\partial}{\partial x} (e^{- 3 x} sin (2 t - 3 x))

- 3 e^{- 3 x} sin (2 t - 3 x) - 3 e^{- 3 x} cos (2 t - 3 x)

Solution steps

\frac{\partial}{\partial x} (e^{- 3 x} sin (2 t - 3 x))

Treat

t

as a constant

Apply the Product Rule:

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

f = e^{- 3 x}, g = sin (2 t - 3 x)

= \frac{\partial}{\partial x} (e^{- 3 x}) sin (2 t - 3 x) + \frac{\partial}{\partial x} (sin (2 t - 3 x)) e^{- 3 x}

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

\frac{\partial}{\partial x} (sin (2 t - 3 x)) = - 3 cos (2 t - 3 x)

= (- 3 e^{- 3 x}) sin (2 t - 3 x) + (- 3 cos (2 t - 3 x)) e^{- 3 x}

Simplify

= - 3 e^{- 3 x} sin (2 t - 3 x) - 3 e^{- 3 x} cos (2 t - 3 x)

What is (\partial)/(\partial x)(e^{-3x}sin(2t-3x)) ?
The answer to (\partial)/(\partial x)(e^{-3x}sin(2t-3x)) is -3e^{-3x}sin(2t-3x)-3e^{-3x}cos(2t-3x)