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To find the derivative of the function \( y = \cos^{-1}(2x) \) using implicit differentiation, we will differentiate both sides with respect to \( x \).

### Step-by-Step Solution:

1. **Start with the given function:**
   \[
   y = \cos^{-1}(2x)
   \]

2. **Apply the cosine inverse function to both sides:**
   \[
   \cos(y) = 2x
   \]

3. **Differentiate both sides implicitly with respect to \( x \):**
   \[
   \frac{d}{dx} [\cos(y)] = \frac{d}{dx} [2x]
   \]

4. **Compute the derivatives:**
   - The derivative of \( \cos(y) \) with respect to \( x \) is \( -\sin(y) \frac{dy}{dx} \).
   - The derivative of \( 2x \) with respect to \( x \) is \( 2 \).

   Thus:
   \[
   -\sin(y) \frac{dy}{dx} = 2
   \]

5. **Solve for \( \frac{dy}{dx} \):**
   \[
   \frac{dy}{dx} = -\frac{2}{\sin(y)}
   \]

6. **Express \( \sin(y) \) in terms of \( x \):**

   Since \( \cos(y) = 2x \), we use the Pythagorean identity:
   \[
   \sin^2(y) = 1 - \cos^2(y) = 1 - (2x)^2 = 1 - 4x^2
   \]

   Thus:
   \[
   \sin(y) = \sqrt{1 - 4x^2}
   \]

   (Note: We choose the positive root because the range of \( \cos^{-1}(x) \) is \( [0, \pi] \), where \( \sin(y) \geq 0 \).)

7. **Substitute \( \sin(y) \) back into the derivative:**
   \[
   \frac{dy}{dx} = -\frac{2}{\sqrt{1 - 4x^2}}
   \]

### Final Answer:
The derivative of the function \( y = \cos^{-1}(2x) \) is:
\[
\frac{dy}{dx} = -\frac{2}{\sqrt{1 - 4x^2}}.
\]

Would you like to know more details or have any questions about the steps?

### Related Questions:
1. What are the derivatives of other inverse trigonometric functions?
2. How do we use implicit differentiation for functions involving both \( x \) and \( y \)?
3. What is the domain of \( y = \cos^{-1}(2x) \)?
4. How can we verify the derivative using the chain rule?
5. What happens if we differentiate with respect to another variable?

### Tip:
Always check the domain of the original function to ensure the derivative is valid in that domain.

Use implicit differentiation to find the derivative of the function y = cos^(-1)(2x).

Learn how to use implicit differentiation to find the derivative of the function y = cos^(-1)(2x). This solution covers important concepts like the derivative of inverse trigonometric functions and the Pythagorean identity. Suitable for calculus students.

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