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Understanding Derivatives: A Chain Rule Approach

Derivatives are a fundamental concept in calculus, allowing us to understand the rate at which functions change. One particularly useful technique in finding derivatives is the Chain Rule. This article provides a comprehensive overview of the Chain Rule through a series of examples, illustrating its application in various mathematical scenarios.

Recap of the Chain Rule

The Chain Rule can be summarized as follows: If we have a function ( y = f(u) ) where ( u ) is itself a function of ( x ) (denoted as ( g(x) )), then the derivative of ( y ) with respect to ( x ) can be expressed as:

[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ]