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Derivatives, the core concept of calculus, originated from Newton’s instantaneous speed, Leibniz’s tangent, and Fermat’s extremum problems, and are defined as the limit of a function’s change ratio, linking continuity, geometry, and physics, while finding extensive applications across science, economics, and biology.
This article explains the definition of matrix‑to‑matrix derivatives, introduces the vectorization‑based differential approach using Kronecker products, presents key matrix‑vectorization properties, and walks through detailed examples illustrating how to compute such derivatives, highlighting their role and limitations in machine‑learning optimization.
This article explains how to compute the derivative of a scalar-valued function with respect to a vector by applying the definition method, illustrating the process with simple and more complex examples, outlining basic differentiation rules, and discussing the limitations of the definition approach for vector‑matrix derivatives.
This article explains the concept of derivatives, covering the rate of change, the limit definition of instantaneous change, geometric interpretation as tangent slopes, and provides derivative formulas and brief proofs for common functions such as constants, linear, quadratic, power, exponential, sine, and cosine.
This article explains the concept of derivatives by introducing the rate of change, defining the instantaneous rate as a limit, interpreting derivatives geometrically as tangent slopes, and presenting the derivatives of common functions such as constants, linear, quadratic, power, exponential, sine, and cosine.