Question

How do you use the second fundamental theorem of Calculus to find the derivative of given `int t^(1/4) dt` from `[1,x]`?

Answer 1

`d/dxint_1^xt^(1/4)dt = x^(1/4)`

Explanation:

The second fundamental theorem of calculus states that if `f` is a continuous function on an open interval `I` and `ainI`, then if
`F(x) = int_a^xf(t)dt`
then
`F'(x) = f(x)` for all `x in I`

As `f(x) = x^(1/4)` is continuous on `(0, oo)` and `1 in (0, oo)`, then if we set
`F(x) = int_1^xf(t)dt`
then, by the second fundamental theorem of calculus,
`F'(x) = f(x) = x^1/4` for all `x in (0, oo)`.

That is to say

`d/dxint_1^xt^(1/4)dt = x^(1/4)`