What is (\partial)/(\partial x)(xye^{x^2+y^2}) ?

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

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(\partial)/(\partial x)(xye^{x^2+y^2})

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\int \frac{2 cos ( x )}{sin ^{2} ( x ) + sin ( x )} d x

x \to 1 lim (\frac{x}{x ^{2} + 2})

f (x) = \frac{3 π x ^{2}}{2}

\int \frac{1 + x}{1 - x ^{2}} d x

\frac{2}{3 x + 1}

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