Calculus: Early Transcendentals, Chapter 5, 5.3, Section 5.3, Problem 14

Hello!
Part 1 of the Fundamental Theorem of Calculus states that for a continuous function f F'_a(x)=f(x), where F_a(x)=int_a^xf(t)dt.

Here f(t)=t^2/(t^4+1) and h(x)=F_1(sqrt(x))
(x>=0).

Therefore
h'(x)=d/(dx)(h(x))=d/(dx)(F_1(sqrt(x)))=F'_1(sqrt(x))*1/(2sqrt(x))=f(sqrt(x))*1/(2sqrt(x))=
=x/(x^2+1)*1/(2sqrt(x))=(sqrt(x))/(2(x^2+1)).