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.