I Why is my derivation of the catenary wrong?

AI Thread Summary
The discussion centers on the differentiation of the tangent function, specifically addressing a common mistake in its approximation. The tangent approximation used, tan(theta + dtheta) ~ tan(theta) + dtheta, is identified as incorrect. The proper differentiation of tan(theta) should involve the quotient rule, leading to a more accurate expression. Participants agree on the need for careful application of differentiation rules in this context. The emphasis is on understanding the correct mathematical approach to avoid errors in solving differential equations.
phantomvommand
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TL;DR Summary
I have "derived" a differential equation for the catenary, and have attached my working. It looks slightly different from the correct expression, which can be found here: https://www.math24.net/equation-catenary

Please do tell me where I made a mistake. Thank you!
Important note: I only derived the differential equation, I did not solve it.

WhatsApp Image 2021-03-04 at 1.22.37 AM.jpeg

What I think caused the mistake:
- the tangent approximation (tan(theta+dtheta) ~ tan theta + d theta
 
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Hi,

phantomvommand said:
What I think

That's what I think too :wink:
The proper way to differentiate ##\tan\theta## is not ##{d\tan\theta\over d\theta} = 1 ## but $${d\tan\theta\over d\theta} = {d\over d\theta}\Biggl ( {\sin\theta\over\cos\theta}\Biggr ) =\ ... $$
​
##\ ##​
 
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