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

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

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

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

3 x^{2} e^{2 x y} + e^{2 x y} \cdot 2 y x^{3}

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x^{3}, g = e^{2 x y}

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

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

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

= 3 x^{2} e^{2 x y} + e^{2 x y} \cdot 2 y x^{3}

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