Part 4/4:
\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1 - x^2}}
]
Conclusion
In closing, we explored how the derivative of the inverse sine function is obtained through implicit differentiation, highlighting key features such as its domain, range, and relationship with cosine. Understanding these principles serves as a foundation not only for (\sin^{-1}(x)) but also paves the way for similar analysis on other inverse trigonometric functions.
Stay tuned for further discussions on derivatives of other inverse trigonometric functions, as the methodologies can be similarly applied across the board. Thank you for engaging in this mathematical exploration!