Proving the Chain Rule Using First Principles: A Comprehensive Guide

Introduction

The chain rule is one of the fundamental principles in calculus, allowing us to find the derivative of a composite function. While proving the chain rule using first principles might seem daunting, we can make use of a few identities and definitions to achieve this. In this article, we will delve into the process of proving the chain rule using the first principle method and provide insights into how to handle various cases.

Understanding the Chain Rule

The chain rule states that if we have a composite function ( (f circ g)(x) ), where ( f ) and ( g ) are both differentiable functions, then the derivative of the composite function is given by:

[ frac{d}{dx}(f circ g)(x) f'(g(x)) cdot g'(x) ]

To prove this using the first principle, we need to start with the definition of the derivative and work through the necessary steps.

Proof Using First Principles

Let's consider the function ( (f circ g)(x) ). According to the definition of the derivative, we have:

[ frac{d}{dx}(f circ g)(x) lim_{h to 0} frac{(f circ g)(x h) - (f circ g)(x)}{h} ]

Substituting the definition of the composite function, we get:

[ frac{d}{dx}(f circ g)(x) lim_{h to 0} frac{f(g(x h)) - f(g(x))}{h} ]

To proceed, we introduce a new variable ( k g(x h) - g(x) ). This allows us to rewrite the expression as:

[ frac{d}{dx}(f circ g)(x) lim_{h to 0} frac{f(g(x h)) - f(g(x))}{g(x h) - g(x)} cdot frac{g(x h) - g(x)}{h} ]

Notice that as ( h to 0 ), ( k to 0 ) as well. We can then separate the limits:

[ frac{d}{dx}(f circ g)(x) lim_{h to 0} frac{f(g(x) k) - f(g(x))}{k} cdot lim_{h to 0} frac{k}{h} ]

Here, ( lim_{h to 0} frac{f(g(x) k) - f(g(x))}{k} ) is the definition of the derivative of ( f ) at ( g(x) ), which is ( f'(g(x)) ). Similarly, ( lim_{h to 0} frac{k}{h} ) can be written in terms of ( g'(x) ) using the mean value theorem or another approach, yielding ( g'(x) ).

[ frac{d}{dx}(f circ g)(x) f'(g(x)) cdot g'(x) ]

This completes the proof using the first principle.

Handling Special Cases

One important case to consider is when ( g(x h) g(x) ). In this scenario, ( k 0 ) and we have to handle the limit carefully. We can use the identity:

[ g(x h) g(x) h cdot g'(x) cdot epsilon_h ]

where ( epsilon_h to 0 ) as ( h to 0 ). This identity helps us to manage the scenario where ( g ) might equal at points other than ( x ).

Conclusion

In conclusion, the chain rule can be proven rigorously using the first principle by employing a series of substitutions and taking limits. This method not only helps in understanding the underlying logic of the chain rule but also ensures that we cover all possible scenarios, including cases where the intermediate function ( g ) might equal the value at ( x ).