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The derivative of a function is the slope of the line that passes through the points where the function's input variables (x) change at a given rate (i.e. increase or decrease by a certain percentage). The derivative can be simplified algebraically to the following equation: increment of x = (increase in x) / (increase in x squared). The derivative of a function can be found by evaluating the limit as the function's input variables approaches zero. In the example shown, the derivative of the function x = 2x is 2x - 1, or 2x = 1. This result can be generalized to other functions by recognizing that the limit of the sum of functions is equivalent to the limit of the individual functions. This knowledge allows us to calculate the slope of a line tangent to a function at any given point.

**00:00:00**In this video, basic geometry concepts will be covered, including the definition of a derivative and its geometric meaning. The focus will be on the recta secante and recta tangente, which are two important geometric curves. The recta secante is a straight line that intersects a circle in two points, while the recta tangente is a straight line that has a common point with a circle. Using this information, a function can be represented as a graph on a Cartesian plane and a recta secante can be drawn that cuts the curve at two points. The recta tangente is also located at the same points as the recta secante, but has a different slope (i.e., the slope of the recta secante is "m", while the slope of the recta tangente is "-"m"). If we only know one point on the curve, using a simple formula for the slope, we can find the slope of the recta tangente. However, if we want to find the slope of a recta tangente that is located on a curve other than the function's graph, we need to use a more sophisticated method. The Greeks over 2000 years ago tackled this problem, and it was revisited in**00:05:00**In this video, you will learn what a derivative is and what it is used for. You will also learn about the different types of derivatives and how to calculate them. Finally, you will see how the derivative of a curve is related to the slope and the tangent of that curve.**00:10:00**The following is a transcript excerpt of a video titled "Y tú, ¿sabes qué es una derivada? Definición y significado geométrico. Cálculo diferencial" followed by a 1-paragraph summary of the application of the limit theorem to find the derivative of a function. After defining a derivative, the video explains that, as the incremental increase of x goes to zero, we would have a single point and we would be back at our original problem. Continuing with the derivation of the mathematical expression that represents the derivative of a function, we can see that the point x2 of 2 goes increasingly close to x1, and 1 will never touch x. We also can see that, as the increments go on, the increase of x tends to be zero, without actually reaching zero. This is because the theory of mathematical limits allows us to find the slopes of different tangents to the function's graph. We can then use the limit theorem to find the derivative of the function itself, which is equal to x squared. Remembering that the derivative is defined as the limit of a function's derivative as x goes to zero, we can replace the term f evaluated at x**00:15:00**In this video, the meaning and definition of a derivative is explained, followed by a discussion of how to calculate a derivative using calculus. The derivative of a function is the slope of the line that passes through the points where the function's input variables (x) change at a given rate (i.e. increase or decrease by a certain percentage). The derivative can be simplified algebraically to the following equation: increment of x = (increase in x) / (increase in x squared). The derivative of a function can be found by evaluating the limit as the function's input variables approaches zero. In the example shown, the derivative of the function x = 2x is 2x - 1, or 2x = 1. This result can be generalized to other functions by recognizing that the limit of the sum of functions is equivalent to the limit of the individual functions. This knowledge allows us to calculate the slope of a line tangent to a function at any given point.

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