Question

How do you find the derivative of `F(x)=cose^tdt` from `[pi, lnx]`?

Answer 1

The Fundamental Theorem of Calculus, part 1, tells us that:

`d/dx (\int _{a}^{x} f(t) \ dt) = f(x)`

But, if we have an interval limit that is a function of `x`, then we need also add on the chain rule:

`d/dx (\int _{a}^{u(x)} f(t) \ dt) = f(u(x)) cdot u'`

So:

`d/dx (int_pi^(ln x) cos(e^t) \ dt )`

`= cose^(ln x) cdot (ln x)'`

`= (cos x)/x`