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Differentiation
Legacy Name: Differentiation

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Differentiation:

finding of the derivative.

Differentiation is a method concerned with the study of rates at which quantities change, i.e., finding the derivative. A derivative is the measure of how a function changes its value as the input changes. Differentiation is the counterpart of integration in calculus.

History:
The modern uses of differentiation were discovered by Newton and Leibniz via the birth of FTC. (See below.)
However, the concepts of antiderivatives date back to Greek geometers like Archimedes and Euclid. Archimedes, specifically, introduced intiniteismals to study areas and volumes.
Newton and Leibniz, considered the godfathers of modern calculus, built on the works of other mathematicians like de Fermat, Barrow, and Descartes. Barrow, in actuality, is credited for the most earliest development of the antiderivative. Nevertheless, Newton and and Leibniz still hold the key to what we know today as calculus, for Leibniz devoloped the notation, where Newton applied differentiation to physics.

The derivative of the function f(x) evaluated at x=a gives the slope of the curve at x=a.

A note: If f is differentiable at a, then f must also be continuous at a.

Derivatives can be donated using:

the Leibniz Notation

Types of derivatives include: Vector-valued functions, partial derivatives, total derivatives, and directional derivatives.

Derivative Basics:

Linear Combination

The Product Rule

The Quotient Rule

The Chain Rule

Fundamental Theorem of Calculus

I literally have a tattoo of this irl fucking fight me.

The fundamental theorem of calculus is a statement that differentiation and integration are inverse operations.
If a continuous function is integrated and differentiated, the original function has been realized. See integration.