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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = x, g = e^{- x - y}

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

\frac{\partial}{\partial x} (x) = 1

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

= sin (y) (1 \cdot e^{- x - y} + (- e^{- x - y}) x)

Simplify

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

4 x^{2} y d x + x y d y = 0

\int_{0}^{5} t e^{- t} d t

\frac{d}{d x} ((f (x))^{4})

f (x) = sin (6 x)

\int \frac{1}{sec ^{2} ( x )} d x

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