What is (\partial)/(\partial x)(x^5e^{xy}) ?

The answer to (\partial)/(\partial x)(x^5e^{xy}) is 5x^4e^{xy}+e^{xy}yx^5

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(\partial)/(\partial x)(x^5e^{xy})

\frac{\partial}{\partial x} (x^{5} e^{x y})

5 x^{4} e^{x y} + e^{x y} y x^{5}

Solution steps

\frac{\partial}{\partial x} (x^{5} e^{x y})

Treat

y

as a constant

Apply the Product Rule:

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

f = x^{5}, g = e^{x y}

= \frac{\partial}{\partial x} (x^{5}) e^{x y} + \frac{\partial}{\partial x} (e^{x y}) x^{5}

\frac{\partial}{\partial x} (x^{5}) = 5 x^{4}

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

= 5 x^{4} e^{x y} + e^{x y} y x^{5}

What is (\partial)/(\partial x)(x^5e^{xy}) ?
The answer to (\partial)/(\partial x)(x^5e^{xy}) is 5x^4e^{xy}+e^{xy}yx^5