What is the correct second derivative for implicit differentiation of r^2 = x^2?

[K41]

I have an equation:

r^2 = x^2

So I found out dr/dx = x/r.

But when I try to find the second derivative, I get d2r/dx2 = -x^2/r^3 when the text says it should be (r^2 - x^2)/r^3.

Can anyone help? My working out:

r^2 - x^2 = 0
r^2 = x^2.
Assume r is a function of x.
rr' = x (first derivative found correctly)
rr'' + r'(x/r) = 1 (apply chain rule and sub in answer for first derivative)
rr'' + x^2/r^2 = 1 (sub in first derivative)

So where have I gone wrong?

 

[Mentallic]

I have an equation:

r^2 = x^2

So I found out dr/dx = x/r.

But when I try to find the second derivative, I get d2r/dx2 = -x^2/r^3 when the text says it should be (r^2 - x^2)/r^3.

Can anyone help? My working out:

r^2 - x^2 = 0
r^2 = x^2.
Assume r is a function of x.
rr' = x (first derivative found correctly)
rr'' + r'(x/r) = 1 (apply chain rule and sub in answer for first derivative)
rr'' + x^2/r^2 = 1 (sub in first derivative)

So where have I gone wrong?

And then where do you go from that last line?

 

[BvU]

Only in the very last steps (after your last line):
rr'' + x²/r² = 1 ⇔
rr'' = 1 - x²/r² ⇔
r'' = 1/r - x²/r³ ⇔
r'' = ( r² - x² ) / r³

 

[K41]

Haha, you won't believe what I was doing. Instead of subtracting both sides, I was doing a division (for reasons not clear to me or anyone of the known realm)...

GGGAAAAAHHHH

Thanks!

 

[BvU]

I believe you. You are not the only one...