We first derive rigorously a partial differential equation (PDE) for speckle reduction from minimizing a cost functional of the instantaneous coefficient of variation. Then, we show that the piecewise exponential function is the solution of the derived PDE. Next, we express the derived PDE using log-compressed ultrasound data, followed by a numerical implementation scheme. Finally, we demonstrate the performance of the proposed PDE using examples and compare the results with those obtained from the speckle reducing anisotropic diffusion (SRAD) algorithm.