Partial derivatives are used in calculus to measure how a function changes as one of its input variables changes, while keeping the other variables constant. This concept is essential when dealing with functions of multiple variables, as it allows us to understand the rate of change in any specific direction. Gradient descent, on the other hand, is an optimization technique often used in machine learning and data science. It involves iteratively adjusting the inputs of a function to find the minimum value by moving in the direction opposite to the gradient, which is a vector consisting of all the partial derivatives. This process helps to minimize errors or losses in various models by efficiently converging towards an optimal solution.