https://file.aiquickdraw.com/mathbot/file/1728869657679gagopsz6.jpg

We are tasked with finding the derivative of the function:

\[
h(t) = \sin^{-1}(2t)
\]

### Step-by-Step Solution:

1. **Recall the derivative of the inverse sine function**:
   The derivative of \( \sin^{-1}(x) \) with respect to \(x\) is:

   \[
   \frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1 - x^2}}
   \]

2. **Chain Rule Application**:
   Since \( h(t) = \sin^{-1}(2t) \), we have an inner function \( g(t) = 2t \). To differentiate, we'll apply the chain rule.

   The chain rule states:
   \[
   \frac{d}{dt} \sin^{-1}(g(t)) = \frac{1}{\sqrt{1 - [g(t)]^2}} \cdot g'(t)
   \]

3. **Differentiate \( g(t) = 2t \)**:
   \[
   g'(t) = 2
   \]

4. **Substitute \( g(t) = 2t \) and \( g'(t) = 2 \) into the chain rule**:
   \[
   h'(t) = \frac{1}{\sqrt{1 - (2t)^2}} \cdot 2
   \]

5. **Simplify**:
   \[
   h'(t) = \frac{2}{\sqrt{1 - 4t^2}}
   \]

### Final Answer:
\[
h'(t) = \frac{2}{\sqrt{1 - 4t^2}}
\]

Let me know if you'd like any further clarification!

### Related Questions:
1. What is the derivative of \( h(t) = \cos^{-1}(3t) \)?
2. How does the chain rule apply to composite trigonometric functions?
3. How do you find the derivative of \( h(t) = \tan^{-1}(5t) \)?
4. What is the domain of the derivative of \( h(t) = \sin^{-1}(2t) \)?
5. How can you differentiate \( h(t) = \sin^{-1}(t^2) \) using the chain rule?

### Tip:
When differentiating inverse trigonometric functions, always remember the domain restrictions of the original function to avoid errors.

Compute the derivative of the function h(t) = sin⁻¹(2t).

Learn how to compute the derivative of the function h(t) = sin⁻¹(2t) using the chain rule. This problem covers differentiation and inverse trigonometric functions, suitable for advanced high school or college calculus students.

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