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\frac{dy}{dx} = -\frac{1}{\sqrt{1 - x^2}}
]
Concluding Thoughts on Inverse Trig Derivatives
This result plays a vital role as it contrasts with the derivative of the inverse sine function, where:
[
\frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}
]
The negative sign in the derivative for inverse cosine indicates that this function decreases as ( x ) increases, while the inverse sine function increases.
In conclusion, the derivative of ( y = \text{cos}^{-1}(x) ) is an important result that can be derived using implicit differentiation and trigonometric identities. This process illustrates not just the utility of calculus in evaluating rates of change, but also the intrinsic connections between trigonometric functions and their inverses.