Part 1/5:
Understanding the Chain Rule Through a Proof
The chain rule is a fundamental concept in calculus, and grasping its proof streamlines diverse derivative applications. In this article, we will dissect the proof of the chain rule, ensuring that the underlying concepts and steps are highlighted for a clear understanding.
Defining the Functions
To begin, we disclose a function defined by a simple equation:
[ f(x) = x^2 + 1 ]
Our objective is to find the derivative ( f'(100) ). However, the complexity arises from the function being self-referential. This function can be reformulated as:
[ f(u) = u^{100} ]
where ( u = x^2 + 1 ). This transformation introduces the concept of a function within a function—a critical aspect of applying the chain rule. We then recognize: