# Implicit Differentiation

1. Oct 5, 2015

### 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?

2. Oct 5, 2015

### Mentallic

And then where do you go from that last line?

3. Oct 5, 2015

### 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

4. Oct 5, 2015

### 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!

5. Oct 5, 2015

### BvU

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