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

The answer to (\partial)/(\partial x)((xy)/(e^{x^2+y^2)}) is e^{-x^2-y^2}y(1-2x^2)

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

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

e^{- x^{2} - y^{2}} y (1 - 2 x^{2})

Solution steps

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

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

= e^{- x^{2} - y^{2}} y (1 - 2 x^{2})

ln (arctan (2 x^{3}))

\int \frac{3}{x} + 2 e^{x} d x

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

\frac{d}{d x} (\frac{4 - x ^{2}}{4 + x ^{2}})

\int (x + \frac{1}{x ^{4} x}) d x

What is (\partial)/(\partial x)((xy)/(e^{x^2+y^2)}) ?
The answer to (\partial)/(\partial x)((xy)/(e^{x^2+y^2)}) is e^{-x^2-y^2}y(1-2x^2)