Due to the pervasiveness of image capturing devices in every-day life, images of individuals are routinely captured. Although this has enabled many benefits, it also infringes on personal privacy. A promising direction in research on obfuscation of facial images has been the work in the k-same family of methods which employ the concept of k-anonymity from database privacy. However, there are a number of deficiencies of k-anonymity that carry over to the k-same methods, detracting from their usefulness in practice. In this paper, we first outline several of these deficiencies and discuss their implications in the context of facial obfuscation. We then develop a framework through which we obtain a formal differentially private guarantee for the obfuscation of facial images in generative machine learning models. Our approach provides a provable privacy guarantee that is not susceptible to the outlined deficiencies of k-same obfuscation and produces photo-realistic obfuscated output. In addition, we demonstrate through experimental comparisons that our approach can achieve comparable utility to k-same obfuscation in terms of preservation of useful features in the images. Furthermore, we propose a method to achieve differential privacy for any image (i.e., without restriction to facial images) through the direct modification of pixel intensities. Although the addition of noise to pixel intensities does not provide the high visual quality obtained via generative machine learning models, it offers greater versatility by eliminating the need for a trained model. We demonstrate that our proposed use of the exponential mechanism in this context is able to provide superior visual quality to pixel-space obfuscation using the Laplace mechanism.