Why the derivative is more than just a slope on a graph
In short: This lecture explains the fundamental shift in perspective required for advanced calculus. By viewing derivatives as linear operators, you can understand how they map small changes in a function's input to corresponding changes in its output, a concept essential for machine learning and beyond.
In the context of MIT's 18.S096 course, Matrix Calculus for Machine Learning and Beyond, instructors Alan Edelman and Steven G. Johnson reframe the derivative. Rather than focusing solely on the geometric slope of a curve, they define the derivative as a linear operator.
This approach allows for the generalization of calculus to complex systems. By analyzing how infinitesimal shifts in input propagate to the output, this framework provides the mathematical foundation necessary for modern computational applications and machine learning algorithms.
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- Summary by Intellect · source: Lecture 1 Part 2: Derivatives as Linear Operators
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