We investigate here the simultaneous approximation of a functionfεCq[−1,1] by a polynomialPnfinterpolatingfand off(1),…f(q)by the respective derivatives ofPnf. The nodes for the interpolation consist ofn“basic” nodes in (−1,1) forn= 1,2,… augmented by“added” nodes for eachn, converging to −1 at the rateandother “added” nodes converging to 1 at the same rate. We show for instance that, ifqis even, in whichLnis the interpolation on the “basic” nodes. Similar results hold ifqis odd. No assumptions need be made about the nodes other than what is already stated. If, for example, one uses for eachnthe zero set of the Chebychev polynomial cos(narc cosx) as the basic nodes, then, and our results thus obtain the rate of convergence which is “best possible.” Similarly, the prescribed rate of convergence to +1 and −1 for the added nodes is the most minimal assumption which gives good approximation of derivatives. In particular, the added nodes can lie partially or completely on top of each other near or at ±1 for some of all values ofnor interspersed between the “basic” nodes with no ill effects.