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

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

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

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

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

Solution steps

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

Simplify

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

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

Treat

y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

\int (x + 2)^{5} d x

\int \frac{1}{3} e^{- \frac{x}{3}} d x

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

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

θ \to - π + lim (csc (θ))

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