To determine which methods can be used to find the derivative of $y = \arcsin x$ with respect to $x$, let’s examine each method proposed:

1. Option I: Using the quotient rule to differentiate $\frac{1}{\sin x}$ is not applicable here. The function $y = \arcsin x$ does not involve $\frac{1}{\sin x}$; it’s defined by the inverse sine function. Therefore, the quotient rule is unnecessary.

2. Option II: Using the chain rule to differentiate $\sin(\arcsin x) = x$ is a valid approach. Since $\sin(\arcsin x) = x$, differentiating both sides with respect to $x$ and applying the chain rule will yield the derivative of $\arcsin x$.

3. Option III: Using implicit differentiation on $\sin y = x$ with respect to $x$ is also a valid approach. If $y = \arcsin x$, then $\sin y = x$. By differentiating both sides with respect to $x$, we can find $\frac{dy}{dx}$ using implicit differentiation.

Conclusion:

Only options II and III are valid methods. Therefore, the correct answer is:

C. II and III only

Would you like more details or have any questions on this solution?

Related Questions:

1. How do you apply the chain rule in differentiating inverse trigonometric functions?
2. What is implicit differentiation, and when is it useful?
3. Can the quotient rule ever be applied in finding derivatives of inverse trigonometric functions?
4. How would you differentiate $y = \arctan x$ using implicit differentiation?
5. How do we derive the general derivative formula for $\arcsin x$?

Tip:

When differentiating functions involving inverse trigonometric functions, implicit differentiation often provides a straightforward solution.