`G(x) = int_1^x(cos(sqrt(t)))dt` Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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 G(x) is the composite of two functions `f(x) = int_1^x cos t dt ` and `g(x) = sqrt x,` hence `G(x) = f(g(x)).`

Since, by FTC, part 1, `f'(x) = cos x` , thenĀ  `G'(x) = f'(g(x))*g'(x).`

`G'(x) = cos(sqrt x)*1/(2sqrt x)`

Hence, evaluating the derivative of the function, using the FTC, part 1, yields `G'(x) = cos(sqrt x)*1/(2sqrt x).`