What is (\partial)/(\partial y)(xye^{-4y}) ?

The answer to (\partial)/(\partial y)(xye^{-4y}) is x(e^{-4y}-4e^{-4y}y)

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(\partial)/(\partial y)(xye^{-4y})

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

x (e^{- 4 y} - 4 e^{- 4 y} y)

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = y, g = e^{- 4 y}

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

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

\frac{\partial}{\partial y} (e^{- 4 y}) = - 4 e^{- 4 y}

= x (1 \cdot e^{- 4 y} + (- 4 e^{- 4 y}) y)

Simplify

= x (e^{- 4 y} - 4 e^{- 4 y} y)

What is (\partial)/(\partial y)(xye^{-4y}) ?
The answer to (\partial)/(\partial y)(xye^{-4y}) is x(e^{-4y}-4e^{-4y}y)