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.