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Numerade Educator

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Problem 28 Medium Difficulty

Find $ f'(x) $ and $ f"(x). $

$ f(x) = \sqrt{x}e^x $

Answer

$e^{x}\left(x^{1 / 2}+\frac{1}{2} x^{-1 / 2}\right)$
$e^{x}\left(x^{1 / 2}+x^{-1 / 2}-\frac{1}{4} x^{-3 / 2}\right)$

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Video Transcript

it's clear is suing you right here. So I'm gonna take the derivative. We're gonna apply the product. Cool. So we have tea over DX X. It's the 1/2 power eats backs Just becomes equal to next to the one have power de over de X e to the X plus each The x d over d x next to the one have this is equal to eat The X Times X to the 1/2 was one have x the negative one have for the second derivative We're gonna apply the product will when we get next. The 1/2 plus one have x to the negative one. Have d over d X for each The X plus e to the x d over DX next to the one have plus 1/2 x to the negative Have we simplify only get hey x Times X tonic 1/2 plus 1/2 x to the negative 1/2 plus e to the x one Have X It's the negative one have plus 1/2 times negative When have X the negative three House This becomes equal to eat The X hands X to the one have less extra negative 1/2 minus 1/4 ex to the negative 3/2