SUMMARY
Arccoth(x) and arctanh(x) do not have the same derivatives due to their differing domains. The derivative of arccoth(x) is given by \(\frac{d}{{dx}}arc\coth (x) = \frac{1}{{1 - x^2 }}\), while the derivative of arctanh(x) is defined only for \(|x| < 1\). The functions themselves are defined as arctanh(x) = \(\frac{1}{2}ln(\frac{1+x}{1-x})\) for \(|x| < 1\) and arccoth(x) = \(\frac{1}{2}ln(\frac{1+x}{x-1})\) for \(|x| > 1\). The disjoint nature of their domains is crucial in understanding their derivatives.
PREREQUISITES
- Understanding of hyperbolic functions, specifically arccoth and arctanh.
- Knowledge of logarithmic differentiation and its applications.
- Familiarity with domain restrictions in mathematical functions.
- Basic calculus concepts, including derivatives and limits.
NEXT STEPS
- Study the properties and applications of hyperbolic functions in calculus.
- Learn about the implications of domain restrictions on function behavior.
- Explore logarithmic differentiation techniques in more depth.
- Investigate the relationship between different inverse hyperbolic functions.
USEFUL FOR
Students and professionals in mathematics, particularly those studying calculus and hyperbolic functions, as well as educators seeking to clarify the differences between arccoth and arctanh derivatives.