How to find 50th derivative of $\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$?

Question

I need to compute 50th derivative of

$$\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$$

Of course I would not compute 50 derivatives. I want to find a certain regularity.

And what I have:

As can be seen, there is a certain regularity. But I can not represent it as a formula. What is the relationship between $105, 945, 10395$?

What do you need to the 50th derivative for? Do you need to evaluate the 50th derivative at a particular value of $x$? — Simon S, Nov 27 '14 at 14:08
Have you considered differentiating the logarithm of the function instead? — Ulrik, Nov 27 '14 at 14:12
$105,945,10395$ are in sequence $A0001147$, the double factorial of odd numbers: $$(2n-1)!! = 1\times 3\times 5\times\cdots\times(2n-1)$$ — Claude Leibovici, Nov 27 '14 at 14:13
@Simon S I have an exercise: Compute the following... $$\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)^{(50)}$$ — martin, Nov 27 '14 at 14:17
If you know (i.e. if you can use) Faa di Bruno's formula for the derivatives of the composition of two functions, you can just write $(\frac{2}{1+x}-1)^{1/2}=f(g(x))$ where $f(x)=\sqrt{x}$ and $g(x)=\frac{2}{1+x}-1$, whose higher derivatives have closed form. — wisefool, Nov 27 '14 at 14:25
Use general Leibniz rule on the product and determine the coefficients by induction. — Not Buying It, Nov 27 '14 at 15:00
Hint: Your function is simply $e^{-\large\text{arctanh }x}$. — Lucian, Nov 27 '14 at 15:29
@Lucian Do we need to convert the degree $−\tanh^{−1}x$? — martin, Nov 27 '14 at 20:38

score 0 · Answer 1 · answered Nov 28 '14 at 19:23

The function is the product of $u=(1-x)^\frac12$ times $v=(1+x)^{-\frac12}.$ If $C(m,n)$ is the binomial coefficient and $P(a,n)$ is the permutation $a(a-1)\cdots (a-n+1)$ [where one defines $P(a,0)=1$] then the $n$th derivative of $u$ is $(-1)^nP(1/2,n)(1-x)^{-n+\frac12},$ while the $n$th derivative of $v$ is $P(-1/2,n)(1+x)^{-n-\frac12}.$ We can then put these into the Leibniz rule to get the $m$th derivative of $u\cdot v$ as the sum of the terms, for nonnegative $r,s$ having sum $m$, given by $$C(m,r)(-1)^rP(1/2,r)P(-1/2,s)(1-x)^{-r+\frac12}(1+x)^{-s-\frac12}.$$ I checked this and it agreed with your calculation for the fourth derivative. However in a way it's not very useful for a really high derivative like the 50th since there would be 51 terms.

How to find 50th derivative of $\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$?

1 Answers1