# 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.