What is (\partial)/(\partial y)(xye^{8y}) ?

The answer to (\partial)/(\partial y)(xye^{8y}) is x(e^{8y}+8e^{8y}y)

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(\partial)/(\partial y)(xye^{8y})

\frac{\partial}{\partial y} (x y e^{8 y})

x (e^{8 y} + 8 e^{8 y} y)

Solution steps

\frac{\partial}{\partial y} (x y e^{8 y})

Treat

x

as a constant

Take the constant out:

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

= x \frac{\partial}{\partial y} (y e^{8 y})

Apply the Product Rule:

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

f = y, g = e^{8 y}

= x (\frac{\partial}{\partial y} (y) e^{8 y} + \frac{\partial}{\partial y} (e^{8 y}) y)

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

\frac{\partial}{\partial y} (e^{8 y}) = e^{8 y} \cdot 8

= x (1 \cdot e^{8 y} + e^{8 y} \cdot 8 y)

Simplify

= x (e^{8 y} + 8 e^{8 y} y)

x \to 2 + lim (\frac{x + 7}{x ^{2} - 4})

x \to 0 lim (\frac{1 - cos ( 9 x )}{x})

f (x) e^{6} + ln (4)

y = \frac{27}{x ^{2} + 9}, at x = 2

\int e^{8 x + 3} d x

What is (\partial)/(\partial y)(xye^{8y}) ?
The answer to (\partial)/(\partial y)(xye^{8y}) is x(e^{8y}+8e^{8y}y)