What is (\partial)/(\partial y)(xye^{9y}) ?

The answer to (\partial)/(\partial y)(xye^{9y}) is x(e^{9y}+9e^{9y}y)

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(\partial)/(\partial y)(xye^{9y})

\frac{\partial}{\partial y} (x y e^{9 y})

x (e^{9 y} + 9 e^{9 y} y)

Solution steps

\frac{\partial}{\partial y} (x y e^{9 y})

Treat

x

as a constant

Take the constant out:

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

= x \frac{\partial}{\partial y} (y e^{9 y})

Apply the Product Rule:

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

f = y, g = e^{9 y}

= x (\frac{\partial}{\partial y} (y) e^{9 y} + \frac{\partial}{\partial y} (e^{9 y}) y)

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

\frac{\partial}{\partial y} (e^{9 y}) = e^{9 y} \cdot 9

= x (1 \cdot e^{9 y} + e^{9 y} \cdot 9 y)

Simplify

= x (e^{9 y} + 9 e^{9 y} y)

y \to + 0 lim (\frac{y}{sin ( 5 y )})

g (x) = x^{\frac{2}{3}}

\int_{0}^{3} t e^{- 1} d t

e^{- x^{2}}

\frac{d}{d x} (\frac{5 x + 1}{x ^{2} - 3})

What is (\partial)/(\partial y)(xye^{9y}) ?
The answer to (\partial)/(\partial y)(xye^{9y}) is x(e^{9y}+9e^{9y}y)