THUWC2017 title.
It does not seem difficult, if a little basic'll derivative is cut. .
And he gave Taylor expansion formula.
Easier.
For the first three formulas seek higher order derivatives.
Guide to 15.16 times on the same subject.
1.$f(x)=sin(ax+b)$
$$D^{(1)}[sin(ax+b)]=acos(ax+b)$$
$$D^{(2)}[sin(ax+b)]=-a^2sin(ax+b)$$
$$D^{(3)}[sin(ax+b)]=-a^3cos(ax+b)$$
$$D^{(4)}[sin(ax+b)]=a^4sin(ax+b)$$
We found regular.
$$D^{(i)}[sin(ax+b)]=\begin{array}a^icos(ax+b)&i=4k+1\\-a^isin(ax+b)&i=4k+2\\-a^isin(ax+b)&i=4k+3\\a^isin(ax+b)&i=4k\end{array}$$
2.$f(x)=e^{ax+b}$
Provided $ g (x) = ax + b, h (x) = e ^ {x} $
$$\begin{array}{rcl}D[f(x)]&=&D[h(g(x))]\\&=&D[h[g(x)]]D[g(x)]\\&=&ae^{ax+b}\end{array}$$
and so
$$D^{(i)}[f(x)]=a^ie^{ax+b}$$
3.$f(x)=ax+b$
$$D^{(i)}[f(x)]=a$$
He also gave Taylor expansion of the equation, can lead directly from zero.
Taylor expansion is:
$$ F (X) = \ SUM \ limits_ I = {0}} ^ {n-\ FRAC {{D ^ (I)} [F (X)] X ^ I {I} $$!}
Then each function into a Taylor series polynomial coefficients calculated for each item, and then use demand factor and direct violence on $ LCT $ chain can be maintained.
Complexity $ (16nlogn) O $