Show that exists on the sphere a derivable curve that connects A and B. Let $A$ and $B$ two points on the unit sphere. Suppose that $A \neq B$. Show that exists on the sphere a derivable curve that connects $A$ and $B$. 
I tried to use the line $R(x)=A+x(B-A)$ with $0 \leq x \leq 1$, 
but I do not know how to start.

You were close, but you chose a straight line segment. You want each point to be a unit distance from the origin. So, provided $A\ne-B$ you should divide each vector by its magnitude.
$$R(x)=\frac{A+x(B-A)}{\Vert A+x(B-A)\Vert}$$
If $A=-B$ the straight line will pass through $(0,0,0)$ and you would be dividing by $0$.
