What is (\partial)/(\partial y)(xye^y) ?

The answer to (\partial)/(\partial y)(xye^y) is x(e^y+e^yy)

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(\partial)/(\partial y)(xye^y)

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = y, g = e^{y}

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

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

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

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

Simplify

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

\int \frac{8 x ^{2} + x + 8}{( x ^{2} + 1 ) ^{2}} d x

f (x) = 2 e^{x}, at x = 2

\int 9 (3 x + 2)^{2} d x

\int sin (22 x) cos (15 x) d x

y = 2 x - x^{2}, y = - x

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