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The trivial thing is to differentiate the expression 5 times and substitute in the formula.

Is there any way to simplify what I wrote in the title out?

Thanks in advance!

user401575
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  • Another option is using wolframalpha, depending on what you're trying to do. If you don't need a formal proof, wolframalpha is your friend –  Mar 08 '17 at 11:55
  • @vrugtehagel Haha unfortuneatly i'm not allowed to use Wolframalpha during the test. I'm looking for ways to simplify the Taylor expansions of certain functions – user401575 Mar 08 '17 at 11:56
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    In that case I'm afraid you'll just have to do it the oldfashioned way, calculating the derivative 5 times. Sometimes, when expressions are a little more complicated like here or here, there's something smart you can do to make it easier, but $\sqrt{x}$ is too basic for that. –  Mar 08 '17 at 12:00

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You can try to find regularities in the derivatives of $\sqrt x$. Notice for $n\geq 2$ the $n$-th derivative of $\sqrt x$ is

$$(-1)^n \frac{(2n-3)!!}{2^n} x^{-\frac{2n-1}2}.$$

I found it by writing down the first three derivatives and by looking closely. I still think that deriving five times may be the faster way. $\sqrt x$ has not so complicated derivatives, especially when written as $x^{\frac12}$.

M. Winter
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