Let f(t)=300(lnt-e^-4t)^3.Calculate f'(1) accurate to three decimal places.
One way to do it is first to calculate the derivative, and then expand the derivative in a power series about some point close to 1. Then use Cauchy's formula for the error term to decide how many terms of the series you need to sum to get that accuracy.
Am getting stuck at the f'(x)
let ln t -e^-4t=u
1/t+4e^-4t dt = du
300(u)^3 . t du/1+4e^-4t
900u^2 . t du/1+4e^-4t
900(ln t -e^-4t)^2 . t du/1+4e^-4t
but this doesn't seem quite right.Any suggestions?