What is (\partial)/(\partial y)(xye^{5y}) ?

The answer to (\partial)/(\partial y)(xye^{5y}) is x(e^{5y}+5e^{5y}y)

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(\partial)/(\partial y)(xye^{5y})

\frac{\partial}{\partial y} (x y e^{5 y})

x (e^{5 y} + 5 e^{5 y} y)

Solution steps

\frac{\partial}{\partial y} (x y e^{5 y})

Treat

x

as a constant

Take the constant out:

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

= x \frac{\partial}{\partial y} (y e^{5 y})

Apply the Product Rule:

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

f = y, g = e^{5 y}

= x (\frac{\partial}{\partial y} (y) e^{5 y} + \frac{\partial}{\partial y} (e^{5 y}) y)

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

\frac{\partial}{\partial y} (e^{5 y}) = e^{5 y} \cdot 5

= x (1 \cdot e^{5 y} + e^{5 y} \cdot 5 y)

Simplify

= x (e^{5 y} + 5 e^{5 y} y)

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What is (\partial)/(\partial y)(xye^{5y}) ?
The answer to (\partial)/(\partial y)(xye^{5y}) is x(e^{5y}+5e^{5y}y)