The derivative of a function with respect to t is denoted as f'(t)
The given function:` f(x) = arcsin(t^2) ` is in a form of a inverse trigonometric function.
Using table of derivatives, we have the basic formula:
`d/(dx)(arcsin(u))= ((du)/(dx))/sqrt(1-u^2)`
By comparison, we may let `u = t^2` then `(du)/(dx)= 2t` .
Applying the formula, we get:
`f'(t) =(2t)/sqrt(1-(t^2)^2)`
`f'(t) =(2t)/sqrt(1-t^4) ` as the first derivative of `f(x)=arcsin(t^2)`
Note: `(t^2)^2 = t^(2*2) = t^4` based on the Law of Exponents: `(x^m)^n = x^(m*n)`
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