how do you find the second derivative of y=e

^{-x^2}how do you find the second derivative of y=e^{-x^2}

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Olney, MD

For the first derivative, use the chain rule:

Let u = -x^{2}

y' = d(e^{u})/du * du/dx = e^{u}*(-2x) = (-2x)e^{-x^2}

For the second derivative use the chain rule and the product rule:

y'' = e^{-x^2}*d(-2x)/dx + (-2x)d(e^{u})/du*(du/dx)

y'' = -2e^{-x^2} + 4x^{2}e^{-x^2}

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

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