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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 = -x2
y' = d(eu)/du * du/dx = eu*(-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(eu)/du*(du/dx)
y'' = -2e-x^2 + 4x2e-x^2
y'' = 2e-x^2(2x2 - 1)