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

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

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(\partial)/(\partial x)(xe^{2xy})

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial x} (x) = 1

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

= 1 \cdot e^{2 x y} + e^{2 x y} \cdot 2 y x

Simplify

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

x y^{2} \frac{d y}{d x} = y^{3} - x^{3}, y (1) = 1

f (x) - 2 x^{2}

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

f (x) = 2 x + 3, (3, 3)

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