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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

x

as a constant

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

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