Non-Renormalization Theorems for Operators with Arbitrary Numbers of Derivatives in $\mathcal{N}=4$ Yang Mills Theory

We generalize the proof of the non-renormalization of the four derivative operators in $\mathcal{N}=4$ Yang Mills theory with gauge group SU(2) to show that certain terms with 2N derivatives are not renormalized in the theory with gauge group SU(N). These terms may be determined exactly by a simple perturbative computation. Similar results hold for finite $\mathcal{N}=2$ theories. We comment on the implications of these results.