`y = int_0^(tan(x))(sqrt(t + sqrt(t)))dt` Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

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You need to use the Part 1 of the FTC to evaluate the derivative of the function. You need to notice that the function y is the composite of two functions `f(x) = int_1^x sqrt(1+sqrt t) dt` and `g(x) = tan x` , hence y = f(g(x)).

Since, by FTC, part 1, `f'(x) = sqrt(1+sqrt x)` , thenĀ  `(dy)/(dx) = f'(g(x))*g'(x)` .

`(dy)/(dx) = sqrt(1+sqrt(tan x))*(1+tan^2 x)`

Hence, evaluating the derivative of the function, using the FTC, part 1, yields `(dy)/(dx) = sqrt(1+sqrt(tan x))*(1+tan^2 x).`