12.3 Summary
This chapter taught us about differentiation, the key component of optimizing functions. Yes, even functions with millions of variables.
Even though we focused on univariate functions (for now), we managed to build a deep understanding of differentiation. For instance, we’ve learned that the derivative
describes the slope of the tangent line drawn to the graph of f at x, which describes the velocity if f is the trajectory of a one-dimensional motion. From the perspective of physics, the derivative describes the rate of change.
However, from the perspective of mathematics, differentiation offers much more than the rate of change: we’ve seen that a differentiable function can be written in the form
around some x0 ∈ℝ. In other words, locally speaking, a differentiable function is a linear part plus a small error term. Unlike the limit-of-quotients definition, this will generalize for multiple variables without an issue. Moreover, we can apply...
