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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

1 \cdot e^{x^{2} y} + e^{x^{2} y} \cdot 2 y xx : e^{x^{2} y} + 2 e^{x^{2} y} x^{2} y

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

f (x) = 3 x^{2} cos (x) + 7 x

\frac{\partial}{\partial x} (x^{2} + x y + 6 x - 6 y)

\frac{d}{d x} (8 - x^{2})

\int \frac{x ^{2}}{x ^{3} - 7} d x

\int_{0}^{4} \frac{1}{x} d x

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