What is (\partial)/(\partial y)(1/(e^{x^2+y^2)}) ?

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

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(\partial)/(\partial y)(1/(e^{x^2+y^2)})

\frac{\partial}{\partial y} (\frac{1}{e ^{x^{2} + y^{2}}})

- 2 e^{- x^{2} - y^{2}} y

Solution steps

\frac{\partial}{\partial y} (\frac{1}{e ^{x^{2} + y^{2}}})

Treat

x

as a constant

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

\int \frac{x ^{3}}{( 2 + 3 x ^{4} ) ^{3}} d x

n = 1 \sum \infty \frac{7}{n}

\frac{\partial}{\partial x} (3 - 3 x^{2} + y^{2})

(8 + x) y^{^{'}} = 5 y

\frac{10}{( s ( s ^{2} + 2 s + 20 ) )}

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