Direct detection of exoplanets is a topic of increasing interest since the first exoplanet discovery by indirect methods. It represents a formidable task because of the small angular separation and large contrast ratio between planet and parent star. We present a novel family of stellar coronagraphs based on the standard coronagraph design but transformed to perform optical differentiation. The proposed coronagraphic masks are used to perform the first or the second derivative of the incoming field. This concept offers a new method to detect exoplanets providing both, deep starlight extinction and high angular resolution. To perform optical differentiation the coronagraph's occulting disk is replaced by an especially designed mask. A further improvement on the coronagraph performance is made by adding a gaussian profile to the differentiation mask in order to reduce the amount of diffracted light. The theoretical rejection rate of our coronagraph is infinite. Computer simulations carried out for the ideal case show that it achieves deep starlight reduction corresponding to a gain of at least 37 mag ($10^{-15}$ light intensity reduction). To take full advantage of the capabilities of our coronagraph atmospheric distortions must be reduced by the use of extreme adaptive optics systems or by its use on space telescopes.