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

The answer to (\partial)/(\partial x)(x^3ye^{xy}) is y(3x^2e^{xy}+e^{xy}yx^3)

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

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

y (3 x^{2} e^{x y} + e^{x y} y x^{3})

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = x^{3}, g = e^{x y}

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

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

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

= y (3 x^{2} e^{x y} + e^{x y} y x^{3})

\frac{\partial}{\partial x} (x^{2} - x + 2 y^{2} + yz - 10)

\frac{\partial}{\partial x} (x (y)^{\frac{1}{2}})

y = \frac{x ^{5}}{10} + \frac{1}{6 x ^{3}}

n = 1 \sum \infty \frac{n}{( n + 1 ) !}

\frac{d}{d t} (4 e^{2 t})

What is (\partial)/(\partial x)(x^3ye^{xy}) ?
The answer to (\partial)/(\partial x)(x^3ye^{xy}) is y(3x^2e^{xy}+e^{xy}yx^3)