Past exam of the mathematical methods II course of the University of Oxford / 2014 / 2 / b / Solution

Expand before taking the fourth derivative:The apparent singularity at zero is removable. The convergent power series may be differentiated termwise, givingAt $x=0.1$, the successive contributions are $-2$, $0.000252$ and approximately $-3.405\times 10^{-9}$. ThusEven the first two nonzero terms give $-1.999748$; the next retained correction quantifies that estimate's accuracy.