To differentiate \( y = \sin^{-1}(x) \) with respect to \( x \), we'll use implicit differentiation. Here’s the step-by-step process:

1. **Start with the equation**: 
   \[
   y = \sin^{-1}(x)
   \]

2. **Take the sine of both sides** to eliminate the inverse sine:
   \[
   \sin(y) = x
   \]

3. **Differentiate both sides** with respect to \( x \), applying the chain rule on the left-hand side:
   \[
   \frac{d}{dx} \left( \sin(y) \right) = \frac{d}{dx}(x)
   \]
   The derivative of \( \sin(y) \) is \( \cos(y) \cdot \frac{dy}{dx} \) (by the chain rule), and the derivative of \( x \) is 1:
   \[
   \cos(y) \cdot \frac{dy}{dx} = 1
   \]

4. **Solve for \( \frac{dy}{dx} \)**:
   \[
   \frac{dy}{dx} = \frac{1}{\cos(y)}
   \]

5. **Express \( \cos(y) \) in terms of \( x \)** using the identity \( \sin^2(y) + \cos^2(y) = 1 \). Since \( \sin(y) = x \), we get:
   \[
   \cos^2(y) = 1 - x^2
   \]
   Therefore, \( \cos(y) = \sqrt{1 - x^2} \) (taking the positive root because the range of \( y = \sin^{-1}(x) \) is \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \)).

6. **Substitute back**:
   \[
   \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}
   \]

Thus, the derivative of \( y = \sin^{-1}(x) \) is:
\[
\frac{d}{dx} \left( \sin^{-1}(x) \right) = \frac{1}{\sqrt{1 - x^2}}
\]

Would you like more details or have any questions about this? 

Here are 5 related questions to explore:
1. How do you differentiate \( y = \cos^{-1}(x) \)?
2. What is the derivative of \( y = \tan^{-1}(x) \)?
3. How can you apply the chain rule to differentiate \( y = \sin^{-1}(2x) \)?
4. What is the range and domain of the function \( y = \sin^{-1}(x) \)?
5. How would you find higher-order derivatives of \( y = \sin^{-1}(x) \)?

**Tip:** Always double-check the domain of inverse trigonometric functions when differentiating. For \( \sin^{-1}(x) \), \( -1 \leq x \leq 1 \).

This page explains how to differentiate y = sin^-1(x) step by step using implicit differentiation and the chain rule. Learn how the derivative is calculated as dy/dx = 1/sqrt(1 - x^2), along with key trigonometric identities and rules.

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