find the derivative of y with respect to the appropriate variable.

y=sin^-1�ã2t

�ã =radical

To find the derivative of y with respect to the variable t, we can use the chain rule.

The given function is y = sin^(-1)(2t).

Let's first rewrite the function as y = arcsin(2t).

Now, let's apply the chain rule to find the derivative:

1. Start with finding the derivative of the outer function, which is arcsin(u). The derivative of arcsin(u) is 1 / sqrt(1 - u^2).

2. Next, find the derivative of the inner function, which is 2t. The derivative of 2t with respect to t is simply 2.

3. Multiply the results from steps 1 and 2 together to find the derivative of the composite function.

Therefore, the derivative of y = arcsin(2t) with respect to t is:

dy/dt = (1 / sqrt(1 - (2t)^2)) * 2

Simplifying further:

dy/dt = 2 / sqrt(1 - 4t^2)

So, the derivative of y with respect to t is 2 / sqrt(1 - 4t^2).