if g(x)=the integral sign (with sin(x^2) on top and 10 at bottom) (1+t) dt, find g'(sqr root pie/2)
Review the formula for differentiating a definite integral.
You end up with three terms:
(1) The integral of the derivative function (in this case, 1) between the same limits.
(2) (PLUS) The value of the integrand at the upper limit, multiplied by the derivative of the upper limit.
(3) (MINUS)The value of the integrand at the lower limit, multiplied by the derivative of the lower limit.
If that sounds too confusing, see
http://mathworld.wolfram.com/LeibnizIntegralRule.html