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

1 Answer
Mar 10, 2016

Answer:

#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)#