Discussion Overview
The discussion revolves around the proof of the derivative of the cosine function. Participants explore various methods of proving this derivative, including geometric definitions, power series, and differential equations, while also considering the potential for shorter proofs.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant presents a detailed proof for the derivative of cosine and seeks feedback on potential errors and shorter methods.
- Another participant suggests that using a geometric definition of sine and cosine necessitates a thorough approach, implying that shortcuts may not be available from that perspective.
- A different approach is proposed using the identity cos(x) = (e^(ix) + e^(-ix))/2, which simplifies the problem to finding the derivative of the exponential function.
- One participant mentions the possibility of applying L'Hôpital's rule to the limit involving sinh, but raises concerns about circular reasoning if the derivative of sine is assumed to be cosine.
- Another participant notes that this question has been raised previously and suggests that a shorter proof exists, referencing the use of angle sum formulas and geometric reasoning to derive the derivatives of sine and cosine.
Areas of Agreement / Disagreement
Participants express differing views on the existence and nature of shorter proofs for the derivative of cosine. While some agree on the correctness of the original proof, others propose alternative methods, indicating that multiple competing views remain without a consensus on the best approach.
Contextual Notes
Some participants highlight the limitations of certain methods, such as the dependence on prior knowledge of derivatives or the potential circularity in using L'Hôpital's rule without establishing the derivatives first. The discussion reflects a variety of assumptions and approaches that may not be universally accepted.