The Chain Rule

The Chain Rule is a fundamental rule in calculus for finding the derivative of composite functions. It states that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function. In mathematical notation, if y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx).

The Chain Rule is particularly useful when dealing with functions that are composed of multiple other functions, such as y = f(g(h(x))). It simplifies the process of finding derivatives by breaking down the composite function into its constituent parts.

For example, let’s consider the function y = sin(x^2). Here, the outer function is f(u) = sin(u), and the inner function is g(x) = x^2. To find the derivative of y with respect to x, we apply the Chain Rule: dy/dx = (dy/du) * (du/dx). In this case, dy/du = cos(u) and du/dx = 2x. Therefore, dy/dx = cos(x^2) * 2x.

The Chain Rule is essential for solving a variety of problems in calculus, including those involving implicit differentiation and related rates.