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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

f (x) = 3 x^{2} - 4 x + 1

\frac{d}{d x} (sin (x) sec (x))

\int_{- 2}^{2} 2 d x

y = 3 x^{2} sin (x)

2 y^{^{''}} + 3 y^{^{'}} - 2 y = 25 e^{- 2 t}

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