Opening discussion of the Chain Rule is here: https://youtu.be/_H3OM2OlqFQ
In this video, we prove the chain rule for multivariable functions. Imagine we have two differentiable functions: f maps from R^n to R^m, and g maps from R^m to R^q. The chain rule tells us that if f is differentiable at a point p and g is differentiable at f(p), then the composition g(f(x)) is differentiable at p as well. Additionally, the Jacobian matrix of gof at p is the product of the Jacobian matrices of g at f(p) and f at p.
The proof focuses on verifying that gof meets the epsilon-delta definition of differentiability using the proposed product of the Jacobians as its associated linear approximation. We begin by defining epsilon and work toward finding a radius delta for which the difference between g(f(x)) and its first-order approximation centered at p is less than epsilon times the distance from x to p. Along the way, we set up bounds to control the behavior of f and g near p and f(p) respectively.
A key step in the proof involves using a "fancy zero" by adding and subtracting a term to help separate out the Jacobians of f and g in the error term. This setup lets us break down the error into two pieces, allowing us to apply known bounds on linear transformations and properties of differentiable functions.
Overall the proof shows two results: (1) the composition gof is differentiable at p, and (2) its Jacobian matrix is exactly the matrix product of the Jacobians of g and f. This confirms the idea that "the Jacobian of a composition is the composition of Jacobians," establishing the higher-dimensional chain rule.
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