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

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

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(\partial)/(\partial x)(e^{xy}+xye^{xy})

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

2 e^{x y} y + e^{x y} x y^{2}

Solution steps

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

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

Simplify

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

= 2 e^{x y} y + e^{x y} x y^{2}

\frac{d y}{d x} = \frac{4 x ^{4} y + 6 y ^{5}}{2 x ^{5} + 3 x y ^{4}}

\int_{\frac{1}{2}}^{3} 6 x ln (2 x) d x

f (x) = 2 x - 3

\frac{d ^{2}}{d x ^{2}} (7 x^{\frac{3}{2}})

\int e^{x} x^{5} d x

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