Find the derivative from x to x^8 (integral) of ln t dt.

Answer 1

f(x) = ∫[x,x^8] ln(t) dt

The 2nd fundamental theorem is really just the chain rule.

f'(x) = ln(x^8)*8x^7 - ln(x)*1/x
= 8lnx*8x^7 - ln * 1/x
= lnx (84x^7 - 1/x)

This works because if

g(t) = ∫lnt dt
g'(t) = ln(t)
So,
f(x) = g(x^8) - g(x)
so, f'(x) = g'(x^8)*8x^7 - g'(x)*1/x
as shown above