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Calculus/derivative of e^(1/2*x)

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Expert: - 3/22/2009


Question
Why is the derivative of e^((1/2)x) 1/2e^((1/2)x)? I thought the rule was f'(e^kx)=(e^kx)/k. Shouldn't the answer be 2e^((1/2)x)? Thank you.

Answer
From the chain rule, we know that the derivative of f(g(x)) is
f'(g(x))g'(x).  Here, f(x) is e^x and g(x) is (1/2)x.

Now the derivative of f(x) = e^x is e^x, so f'(x) = f(x) and
the derivative of (1/2)x is 1/2.

This makes f'(g(x))g'(x) be [e^(x/2)]((1/2)x).

Note that (1/2)x is the same as x/2.

Writing this in it's proper form gives xe^(x/2)/2.  

Calculus

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