| It is well-known that the Immersed Interface Method (IIM) is second order accurate for interface problems. But the accuracy of the first order derivatives, or gradients for short, is not so clear and is often assumed to be first order accurate. In this report, new strategies based on IIM are proposed for elliptic interface problems to compute the gradient at grid points both regular and irregular, and at the interface from each side of the interface. Second order in 1D, or nearly second order (except a factor of $ | \log h | $) convergence in 2D of the computed gradient is obtained with almost no extra cost, and has been explained in intuition and verified by non-trivial numerical tests. |