SUMMARY
The derivative of the inverse hyperbolic cosine function, denoted as cosh-1x, is confirmed to be \(\frac{1}{\sqrt{x^2 - 1}}\). The inverse function is expressed as \(ln(x + \sqrt{x^2 - 1})\). To derive this, one can set \(y = arccoshx\), apply the definition \(coshy = x\), and differentiate using the identity \(cosh^2(y) - sinh^2(y) = 1\). This method validates the derivative effectively.
PREREQUISITES
- Understanding of hyperbolic functions, specifically cosh and sinh.
- Familiarity with inverse functions and their derivatives.
- Knowledge of logarithmic functions and their properties.
- Basic calculus concepts, including differentiation techniques.
NEXT STEPS
- Study the properties and graphs of hyperbolic functions.
- Learn about the derivation of inverse trigonometric functions.
- Explore the applications of hyperbolic functions in calculus.
- Investigate the relationship between hyperbolic and circular functions.
USEFUL FOR
Students studying calculus, mathematics educators, and anyone interested in advanced mathematical concepts related to hyperbolic functions and their derivatives.