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

The answer to (\partial)/(\partial x)(x^2ye^{x^2+y^2}) is y(2e^{x^2+y^2}x+2e^{x^2+y^2}x^3)

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

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

y (2 e^{x^{2} + y^{2}} x + 2 e^{x^{2} + y^{2}} x^{3})

Solution steps

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

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

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

e^{x^{2} + y^{2}} \cdot 2 x x^{2} = 2 e^{x^{2} + y^{2}} x^{3}

= y (2 e^{x^{2} + y^{2}} x + 2 e^{x^{2} + y^{2}} x^{3})

\int \frac{x - 1}{x ^{5}} d x

\int \frac{x ^{2}}{x ^{2} - 6 x + 13} d x

(1 + x^{4}) d y = - x (1 + 4 y^{2}) d x

x \to 0 lim (\frac{1}{x} \cdot sin (x))

y^{^{'}} = x y

What is (\partial)/(\partial x)(x^2ye^{x^2+y^2}) ?
The answer to (\partial)/(\partial x)(x^2ye^{x^2+y^2}) is y(2e^{x^2+y^2}x+2e^{x^2+y^2}x^3)