What is (\partial)/(\partial x)(x^2y^3e^{xy}) ?

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

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(\partial)/(\partial x)(x^2y^3e^{xy})

\frac{\partial}{\partial x} (x^{2} y^{3} e^{x y})

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

Solution steps

\frac{\partial}{\partial x} (x^{2} y^{3} e^{x y})

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = x^{2}, g = e^{x y}

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

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

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

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

x \to \infty lim (\frac{9 x - x ^{2} - 7 x ^{4}}{x ^{3} + 12 x})

\int \frac{( x ^{6} e ^{x} )}{x} d x

\int (1 - x) x^{3} (x - 2)^{2} d x

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

\frac{d}{d x} (5 x^{0.5})

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