Question

Use Part 1 of the FTC to find the derivative of each of the following functions:
(a.) g(x)=integral (from 1 to x) ln(t)dt
(b) F(x)=integral (from x to 10) tanthetadtheta
(c)h(x)=integral (from 0 to x^2) (1+sqrt(r))dr

Answer 1

ln(x)

put a negative in front to get - integral from 10 to x of tan theta d theta, to get -tanx

substitute u =x^2 for the last one, gtg

Answer 2

FTC\[\int_{a(x)}^{b(x)}f(t)~dt=F(b(x))-F(a(x))\]
taking the derivative we get
\[D_x[F(b(x))-F(a(x))]=f(b(x))b'(x)-f(a(x))a'(x)\]

Answer 3

or to cut the clutter
\[\frac d{dx}~\int^{b}_{a}f(t)~dt=f(b)b'-f(a)a'\]