What is the derivative of #e^((x^2)/2)#?

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Answer 1 · 2016-02-22T00:49:46

Use the chain rule to find that

#d/dx e^(x^2/2) = xe^(x^2/2)#

The chain rule states that given two differentiable functions #f# and #g#

#d/dxf@g(x) = f'(g(x))*g'(x)#

In this case, let #f(x) = e^x# and #g(x) = 1/2x^2#

Then #f'(x) = e^x# and #g'(x) = 1/2(2x) = x#

So, by the chain rule, we have

#d/dxe^(x^2/2) = d/dxf@g(x)#

#=f'(g(x))*g'(x)#

#=e^(g(x))*x#

#=xe^(x^2/2)#