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

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Discussion Overview

The discussion revolves around finding the correct second derivative for the implicit differentiation of the equation r2 = x2. Participants explore the derivation process, including the first and second derivatives, and seek clarification on discrepancies between their results and those presented in a text.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant finds the first derivative dr/dx = x/r and attempts to derive the second derivative, leading to d2r/dx2 = -x2/r3.
  • Another participant suggests that the correct second derivative should be (r2 - x2)/r3 and provides a step-by-step transformation of the equation to support this claim.
  • One participant acknowledges a mistake in their approach, indicating confusion over the algebraic manipulation involved in reaching the correct form of the second derivative.
  • Another participant expresses solidarity with the original poster's confusion, suggesting that such errors are common in mathematical reasoning.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct form of the second derivative, as there are competing views on the derivation process and the resulting expressions.

Contextual Notes

Participants express uncertainty regarding specific algebraic steps and the implications of their manipulations, indicating that assumptions about the relationships between variables may not be fully resolved.

K41
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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?
 
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djpailo said:
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?
 
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
 
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!
 
I believe you. You are not the only one...
 
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