I'm trying to find the Taylor expansion of sqrt(x) of order 5 in the point 1 in a simple way

Question

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!

Answer 1

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 $x^{\frac12}$.