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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

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