👉 Reglas de Derivación Básicas 💥TRUCOS Tablas Derivadas Ejemplos y Ejercicios resueltos División ...

Summary

The video presents a practical review of four foundational differentiation rules: differentiating polynomials, products, quotients, and powers of functions. The instructor emphasizes that later derivative techniques build on these basics, so it is important to learn the rules and apply them carefully.

1. Differentiating polynomials

• A constant differentiates to zero: [ \frac{d}{dx}(c)=0 ]

• A constant times (x) differentiates to the constant: [ \frac{d}{dx}(ax)=a ]

• Power rule: Bring the exponent down as a multiplier, then reduce the exponent by one: [ \frac{d}{dx}(x^n)=nx^{n-1} ]

• Differentiate each term of a polynomial separately, keeping its sign and coefficient. Then combine any like terms.

• The instructor recommends writing out the steps while learning, even if zero terms or factors of (1) can later be omitted.

2. Product rule

For two functions multiplied together, use: [ (fg)’=f’g+fg’ ]

In words: differentiate the first factor and multiply by the second unchanged, then add the first unchanged times the derivative of the second.

The examples apply this rule to products of polynomials. Keeping the two parts in parentheses helps preserve the structure and avoid mistakes.

3. Quotient rule

For a function divided by another function: [ \left(\frac{f}{g}\right)’=\frac{f’g-fg’}{g^2} ]

A step-by-step approach is:

• Differentiate the numerator and multiply by the denominator unchanged.
• Subtract the numerator unchanged multiplied by the derivative of the denominator.
• Divide the entire result by the denominator squared.
• Keep the subtraction in parentheses and pay close attention to negative signs.
• Do not cancel terms across addition or subtraction. Cancellation is valid only for common factors that multiply the entire numerator and denominator.
• The denominator is often left squared rather than expanded, since that can make the expression easier to use later.

The instructor repeatedly identifies sign errors and invalid cancellation as common mistakes.

4. Power rule for a function inside a power

When a function is raised to a numerical power, apply the power rule and then multiply by the derivative of the inner function: [ \frac{d}{dx}\big(u(x)^n\big)=n\,u(x)^{n-1}u’(x) ]

The key is to remember the inner derivative (u’(x)). The examples show this rule being combined with polynomial differentiation.

General advice

• Practise the rules until applying them becomes routine.
• Say or write the rule as you work to help remember its sequence.
• Check signs, exponents, and parentheses before simplifying.
• Treat each new rule as building on earlier ones. The video previews that later material will combine these rules and introduce additional derivative topics.

Speakers and sources

• Sergio is the instructor and presenter, associated with the profesor10demates channel.
• No other speakers are clearly featured in the subtitles.