The first few terms of the Maclaurin series can be obtained explicitly:
$$1-x-{\frac { \left( d-1 \right) \left( K-1 \right) }{2\,Kd+2}}{x}^{2 }-{\frac { \left( d-1 \right) \left( d-2 \right) \left( K-1 \right) \left( K-2 \right) }{ \left( 6\,Kd+6 \right) \left( Kd+2 \right) }}{ x}^{3}+O \left( {x}^{4} \right) $$
The coefficient of $x^2$ is not especially small, but of course on a small interval around $0$ the $1-x$ dominates. I guess the question is really to estimate $h''(x)$ on the interval $[0,.1]$.