Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 2 / 9 / ii / Solution

Let $g(x)=\tan (x+\pi /4)$. The derivative identity for tangent gives $g^{\prime }=1+g^2$, so $g(0)=1$, $g^{\prime }(0)=2$, and $g^{\prime \prime }(0)=2g(0)g^{\prime }(0)=4$. Its first three nonzero Taylor expansion terms are thereforeThe factor $1/2!$ on the second derivative is what turns $g^{\prime \prime }(0)=4$ into the coefficient $2$.