Explanation of the derivative by definition of a polynomial function

Greetings friends of the steem platform, especially to the WORLD OF XPILAR community.

On this occasion I continue to share with all of you my plan of action to keep my calculus II and calculus III students active in the teaching and learning of derivatives, integrals and graphs of polynomial functions in the Food Engineering program at the Universidad Experimental Sur del Lago Jesús María Semprum (UNESUR).

In this case I provided my calculus II students with exactly the demonstration and deduction of how the derivative equation was arrived at by definition, that is why to complement that deduction I thought of the idea of solving an exercise where the derivative of a real function has to be found by applying the definition.

Methodology and application of the step-by-step approach to solve the derivative by definition of the function

1] From the function that we want to derive using as a tool the derivative by definition, we must bear in mind that where the variable x exists, x+∆x is replaced.

2] From the equation of the derivative by definition, when we are told that we must subtract f(x), the first thing is to open a bracket or parenthesis in order to know that the following minus multiplies all the terms of f(x)

3] If in the third step there is any mathematical property or mathematical artifice to be applied, it must be applied taking into account that one must be aware of any artifice that can be applied, among these are

Remarkable product.
Rationalization.
Conjugated.
Distributive property.
Empowerment.
Radiation.

4] Once any mathematical artifice has been applied, the terms that are equal and with different signs are simplified, that is, the algebraic sum of terms is performed.

5] In this step, after having simplified the terms, we should have a sum of terms where the common factor is ∆x, so we should take out the common factor ∆x which has the lesser exponent, and leave within the parenthesis the terms that complete the exponent of the original terms.

6] In this step the terms ∆x are cancelled, both the one in the numerator and the denominator.

7] As a last step and having taken the common factor ∆x and simplified the ∆x of the numerator with that of the denominator, then finally we can apply the limit of when the delta of x tends to be zero of the expression that we have left, the result that we have left there finally will be the resulting expression that is the derivative of the real function.

Contribution and possible learning left by the approach of this exercise of derivative by definition

During the time in which Leibniz and Newton worked separately to contribute this great jewel of differential calculus, the first thing that was clearly achieved was the derivative by definition. However, the existing complexity to achieve the derivative of slightly more complex functions made it evolve to find certain basic rules of derivation (formulas) that would be tabulated and demonstrated using the derivative by definition as a support and basis.

That is why after my students have learned to derive certain functions from real variables by definition, it is necessary to make comparisons of the derivatives solved by definition and compare their resolution in which they are solved by the basic rules of derivation, which facilitate the derivation process in comparison to those derived by definition.

---

Note: All images are my property, the cover image was created using Microsoft PowerPoint design tools.