SUMMARY
The derivative of the function f(x) = sinh(x)/cosh(x) is correctly calculated using the quotient rule, yielding f'(x) = sech²(x). The discussion emphasizes the importance of applying the quotient rule accurately and recognizing hyperbolic identities, specifically that cosh²(x) - sinh²(x) = 1. Participants clarified that sech(x) is equivalent to 1/cosh(x), leading to the final simplified result of sech²(x) for the derivative.
PREREQUISITES
- Understanding of hyperbolic functions, specifically sinh(x) and cosh(x).
- Familiarity with the quotient rule for derivatives.
- Knowledge of hyperbolic identities, particularly cosh²(x) - sinh²(x) = 1.
- Basic algebraic manipulation skills for simplifying expressions.
NEXT STEPS
- Study the quotient rule in detail, including examples and common pitfalls.
- Learn about hyperbolic function identities and their applications in calculus.
- Practice finding derivatives of various hyperbolic functions.
- Explore the relationship between hyperbolic and trigonometric functions.
USEFUL FOR
Students studying calculus, particularly those focusing on derivatives of hyperbolic functions, as well as educators seeking to clarify concepts related to the quotient rule and hyperbolic identities.