What is (\partial)/(\partial y)(x^3ye^{xy}) ?

The answer to (\partial)/(\partial y)(x^3ye^{xy}) is x^3(e^{xy}+e^{xy}xy)

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(\partial)/(\partial y)(x^3ye^{xy})

\frac{\partial}{\partial y} (x^{3} y e^{x y})

x^{3} (e^{x y} + e^{x y} x y)

Solution steps

\frac{\partial}{\partial y} (x^{3} y e^{x y})

Treat

x

as a constant

Take the constant out:

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

= x^{3} \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^{3} (\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^{3} (1 \cdot e^{x y} + e^{x y} x y)

Simplify

= x^{3} (e^{x y} + e^{x y} x y)

y = \frac{7}{x ^{2} + 3}

\frac{\partial}{\partial x} (- y - \frac{x}{y ^{2}})

\int \frac{3}{x x ^{2} - 9} d x

\int e^{- x^{4}} (- 4 x^{3}) d x

\int (5 x^{4} + 2) (x^{5} + 2 x)^{2021} d x

What is (\partial)/(\partial y)(x^3ye^{xy}) ?
The answer to (\partial)/(\partial y)(x^3ye^{xy}) is x^3(e^{xy}+e^{xy}xy)