`F(x) = int_sqrt(x)^(2x)(arctan(t))dt` Find the derivative of the function.

Textbook Question

Chapter 5, 5.3 - Problem 58 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Hello!

Let's temporarily denote the antiderivative of `arctan(t)` as `A(t).`

Then `F(x)= A(2x)-A(sqrt(x)),`

and

`F'(x)=A'(2x)*2-A'(sqrt(x))*(1/(2sqrt(x))).`

Recall what `A'(x)` is and obtain

`F'(x)=2arctan(2x)-(arctan(sqrt(x)))/(2sqrt(x)).`

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