DERIVADA DE UNA FUNCIÓN USANDO LA DEFINICIÓN - Ejercicio 2

Summary

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Takeaways

• 😀 The derivative of a function is defined as the limit of the difference quotient: (f(x + Δx) - f(x)) / Δx as Δx approaches 0.
• 😀 The definition of the derivative can use either Δx or h as the change in x, but here we use Δx.
• 😀 To evaluate the derivative, the function f(x) is expanded by substituting (x + Δx) and applying algebraic expansions.
• 😀 Algebraic skills are essential when working with derivatives, especially when expanding binomials and applying notable product formulas.
• 😀 A binomial squared is expanded using the formula: (a + b)^2 = a^2 + 2ab + b^2.
• 😀 The cube of a binomial is expanded using the formula: (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.
• 😀 Distributive property is used to multiply constants through the expanded expressions, simplifying the equation.
• 😀 Terms that cancel each other out (e.g., positive and negative versions of the same term) can be eliminated to simplify the expression.
• 😀 The limit form for the derivative produces an indeterminate form (0/0), which is resolved by factoring out Δx from the numerator.
• 😀 After factoring out Δx, the limit is evaluated as Δx approaches 0, eliminating the indeterminate form and simplifying the derivative expression.
• 😀 The final result for the derivative of the function is: f'(x) = 10x - 21x^2, computed using the limit definition of the derivative.

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