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This paper presents a new method to perform efficient-first-order symbolic sensitivity analysis of analog circuits by direct differentiation of symbolic expressions stored as element-coefficient diagrams (ECDs). An ECD is a compact graphical representation of a symbolic transfer function. It is the cancellation-free and per-coefficient term generation version of determinant decision diagrams (DDDs). The symbolic sensitivity equations obtained from ECDs are stored as a sensitivity-ECDs (SECDs) and can be evaluated extremely fast as it inherits the properties of ECDs. The proposed methodology has been applied to the calculation of sensitivities of four benchmark circuits and it has been demonstrated to be as accurate and more efficient than numerical sensitivity analysis done by SPECTRE.