Question

How do you find the derivative of `f(x) =intcos 2t dt` over `[x,pi/4]`?

Answer 1

So,

`f(x) = int_x^(pi/4) cos(2t) dt = - int_(pi/4)^x cos(2t) dt`.

The fundamental theorem says that `f'(x) = - cos(2x)`.

Theorem (fundamental). Let `f` be a continuous function over an interval `I`. If `a in I` is a constant, the function defined by
`F(x) = int_a^x f(t) dt`
is differentiable over `I`, and `F'(x) = f(x)` for all `x in I`.