What is (\partial)/(\partial x)(ce^{-x}sin(4x)) ?

The answer to (\partial)/(\partial x)(ce^{-x}sin(4x)) is c(-e^{-x}sin(4x)+4e^{-x}cos(4x))

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(\partial)/(\partial x)(ce^{-x}sin(4x))

\frac{\partial}{\partial x} (c e^{- x} sin (4 x))

c (- e^{- x} sin (4 x) + 4 e^{- x} cos (4 x))

Solution steps

\frac{\partial}{\partial x} (c e^{- x} sin (4 x))

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= c \frac{\partial}{\partial x} (e^{- x} sin (4 x))

Apply the Product Rule:

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

f = e^{- x}, g = sin (4 x)

= c (\frac{\partial}{\partial x} (e^{- x}) sin (4 x) + \frac{\partial}{\partial x} (sin (4 x)) e^{- x})

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

\frac{\partial}{\partial x} (sin (4 x)) = cos (4 x) \cdot 4

= c ((- e^{- x}) sin (4 x) + cos (4 x) \cdot 4 e^{- x})

Simplify

= c (- e^{- x} sin (4 x) + 4 e^{- x} cos (4 x))

What is (\partial)/(\partial x)(ce^{-x}sin(4x)) ?
The answer to (\partial)/(\partial x)(ce^{-x}sin(4x)) is c(-e^{-x}sin(4x)+4e^{-x}cos(4x))