Partial derivatives of implicit functions arise when the relationship between variables is defined implicitly, rendering it difficult to solve for one variable explicitly. The Implicit Function Theorem provides a framework for finding the partial derivatives of such functions. By differentiating the implicit equation implicitly with respect to one variable while treating the other as a constant, we can express the partial derivative as a quotient of partial derivatives of the implicitly defined function. This technique is crucial for analyzing implicit relationships and understanding the behavior of functions in multivariable spaces.
