Second Derivative of Time Dilation Equation

[Eismcsquared]

Hello all.
I was playing around with the time dilation equation : √(1-v²/c²)
Specifically, I decided to take the derivative(d/dv) of the equation. Following the rules of calculus, as little of them as I know, I got this:
d/dv(√(1-v²/c²) = v / (c²√(1-v²/c²)).
Now, this seems reasonable enough, and when I (don’t hate me for this) plugged in our original equation into Desmos, and asked for the derivative as well, my equation seemed to be correct.
However, every time I attempt to produce the second derivative of the equation, I end up with an output of simply -1. Naturally this is wrong, and desmos agrees.
Here’s my work:
-v • d/dv(1/√(1-v²/c²))•d/dv(-v)•1/√(1-v²/c²)
Continuing...
-v•1/(-v / c²√(1-v²/c²))
• 1/c²√(1-b²/c²) • -1
Simplifying...
-v•c²√(1-v²/c²)/(-v)
• -1/(c²√(1-v²/c²)
Simplifying some more, i get -1??
I have no clue where I’m going wrong.. I think it’s simply a matter of me not quite understanding one the calculus rules I used in this.
I’m sorry for any formatting errors/messiness but I’ll try to clean it up as best I can. I can’t seem to find the fraction button.

 

[PeterDonis]

I got this

Note that you left out the minus sign in front here. You put it in in your work further down, but I wanted to note that it's left out here.

Here’s my work

You're using the chain rule, which is fine. I would advise using units where $c = 1$, which will reduce clutter and make it easier to do the math. That gives this for the first line of your work:

$$
\frac{d}{dv} \left ( - v \frac{1}{\sqrt{1 - v^2}} \right) = \left( \frac{d}{dv} - v \right) \left( \frac{1}{\sqrt{1 - v^2}} \right) - v \frac{d}{dv} \left( \frac{1}{\sqrt{1 - v^2}} \right)
$$

However, I don't think you're performing the derivative of $1 / \sqrt{1 - v^2}$ correctly. It might help to rewrite it using explicitly exponents, so that it's easier to take derivatives using the power rule $d (x^n) = n x^{n - 1}$. Rewriting the second term above this way, and evaluating the first term since it's simple, you should get this:

$$
- \left( 1 - v^2 \right)^{- 1/2} - v \frac{d}{dv} \left( 1 - v^2 \right)^{- 1/2}
$$

Does that help to see how to correctly take the derivative of the second term?

 

[Eismcsquared]

Yes, thank you!