Find the derivative of the function.

*y* = sin−1(cos−1(*t*))

To find the derivative of the function:

\[
y = \sin^{-1}(\cos^{-1}(t))
\]

we can proceed with the following steps:

1. **Differentiate with respect to \( t \)**:
   Since \( y = \sin^{-1}(\cos^{-1}(t)) \), we apply the chain rule.

   \[
   \frac{dy}{dt} = \frac{d}{dt} \sin^{-1}(\cos^{-1}(t)) = \frac{1}{\sqrt{1 - (\cos^{-1}(t))^2}} \cdot \frac{d}{dt} (\cos^{-1}(t))
   \]

2. **Differentiate the inner function**:
   We know that the derivative of \( \cos^{-1}(t) \) with respect to \( t \) is:
   
   \[
   \frac{d}{dt} (\cos^{-1}(t)) = -\frac{1}{\sqrt{1 - t^2}}
   \]

   So we substitute this result:

   \[
   \frac{dy}{dt} = \frac{1}{\sqrt{1 - (\cos^{-1}(t))^2}} \cdot \left(-\frac{1}{\sqrt{1 - t^2}}\right)
   \]

3. **Simplify the expression**:
   The expression becomes:

   \[
   \frac{dy}{dt} = -\frac{1}{\sqrt{(1 - (\cos^{-1}(t))^2)(1 - t^2)}}
   \]

This is the derivative of the function \( y = \sin^{-1}(\cos^{-1}(t)) \) with respect to \( t \).

Would you like further explanation on each step or help with another problem?

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Here are some related questions to practice:

1. What is the derivative of \( y = \cos^{-1}(\sin^{-1}(t)) \)?
2. How would the process change if \( y = \tan^{-1}(\cos^{-1}(t)) \)?
3. Find the derivative of \( y = \sin(\cos^{-1}(t)) \).
4. What are the derivatives of \( y = \cos^{-1}(x) \) and \( y = \sin^{-1}(x) \) individually?
5. How can we simplify expressions involving inverse trigonometric functions in derivatives?

**Tip:** Remember, using the chain rule effectively is key for differentiating nested inverse trigonometric functions.

Find the derivative of the function y = sin−1(cos−1(t))

Learn how to differentiate the function y = sin−1(cos−1(t)) step-by-step. This calculus problem covers inverse trigonometric functions and the chain rule, offering detailed guidance on each part of the solution. Ideal for advanced high school or college-level students studying calculus.

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