Gauge invariance in both quantum and classical mechanics is investigated using the Feynman path integral formulation of quantum mechanics. The Feynman path integral expresses the wave function in terms of the Lagrangian. A gauge transformation in classical mechanics is the addition of a total derivative of a gauge function to the Lagrangian. A gauge transformation of the second and first kind in quantum mechanics is obtained by adding and subtracting, respectively, a total derivative to the Lagrangian in the Feynman path integral.