how do you find the second derivative of y=e^{-x^2}
Differentiation Rules for Exponential Functions
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How do you find the second derivative of y=e^{-x^2}
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)