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

$f \left(x\right) = {\int}_{x}^{\frac{\pi}{4}} \cos \left(2 t\right) \mathrm{dt} = - {\int}_{\frac{\pi}{4}}^{x} \cos \left(2 t\right) \mathrm{dt}$.
The fundamental theorem says that $f ' \left(x\right) = - \cos \left(2 x\right)$.
Theorem (fundamental). Let $f$ be a continuous function over an interval $I$. If $a \in I$ is a constant, the function defined by
$F \left(x\right) = {\int}_{a}^{x} f \left(t\right) \mathrm{dt}$
is differentiable over $I$, and $F ' \left(x\right) = f \left(x\right)$ for all $x \in I$.