What is (\partial)/(\partial x)(e^{-x}sin(x+y)) ?

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

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(\partial)/(\partial x)(e^{-x}sin(x+y))

\frac{\partial}{\partial x} (e^{- x} sin (x + y))

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

Solution steps

\frac{\partial}{\partial x} (e^{- x} sin (x + y))

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

\frac{\partial}{\partial x} (sin (x + y)) = cos (x + y)

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

Simplify

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

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