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).