To determine the domain of the function \( f(x) = \cos^{-1}(2x) \), we need to consider the range of values for which the expression inside the inverse cosine function is valid. The primary condition for the inverse cosine function \( \cos^{-1}(y) \) is that its input \( y \) must be within the interval \([-1, 1]\).
Here, the input is \( 2x \), so we set up the inequality:
\(-1 \leq 2x \leq 1\)
We solve this compound inequality for \( x \) by dividing all parts of the inequality by 2:
\(-\frac{1}{2} \leq x \leq \frac{1}{2}\)
Thus, the domain of the function \( f(x) = \cos^{-1}(2x) \) is the interval:
\(\left[-\frac{1}{2}, \frac{1}{2}\right]\)
For the curve \( \sqrt{x} + \sqrt{y} = 1 \), find the value of \( \frac{dy}{dx} \) at the point \( \left(\frac{1}{9}, \frac{1}{9}\right) \).