Part 5/5:
[
\frac{d}{dx}(\operatorname{arccosh}(x)) = \frac{1}{\sqrt{x^2 - 1}}, \quad x \geq 1
]
- Derivative of ( \operatorname{arctanh}(x) ):
[
\frac{d}{dx}(\operatorname{arctanh}(x)) = \frac{1}{1 - x^2}, \quad |x| < 1
]
Conclusion
Understanding these derivatives not only aids in calculus but also enhances comprehension of mathematical functions' behaviors. Remembering the domain restrictions and using implicit differentiation as a strategy can make deriving these functions much more manageable. As demonstrated, the interconnectedness of hyperbolic functions with their inverses provides a rich area of study in mathematics. Stay tuned for further explorations in mathematical derivatives and functions!