DERIVADAS TRIGONOMETRICAS Y REGLA DE LA CADENA ONCE M2 S23 FINAL

Summary

The lesson introduces trigonometric derivatives, reviews basic differentiation rules, and explains how to use the chain rule for composite functions.

1. Trigonometric Derivative Rules

The instructor presents derivatives using (f’(x)) notation and the equivalent “derivative with respect to (x)” notation. The standard formulas are:

• (\frac{d}{dx}(\sin x)=\cos x)
• (\frac{d}{dx}(\cos x)=-\sin x)
• (\frac{d}{dx}(\tan x)=\sec^2 x)
• (\frac{d}{dx}(\cot x)=-\csc^2 x)
• (\frac{d}{dx}(\sec x)=\sec x\tan x)
• (\frac{d}{dx}(\csc x)=-\csc x\cot x)

Note: Some signs in the auto-generated subtitles appear to be missing. The formulas above give the standard derivatives, including the negative signs for cotangent and cosecant.

2. Sums, Products, and Quotients

The lesson applies familiar differentiation rules to expressions involving trigonometric functions:

• Sum rule: Differentiate each term separately. For example, differentiating (\sin x+\cos x) gives (\cos x-\sin x).

• Product rule: For (u(x)v(x)), [ (uv)’=u’v+uv’. ] The example involving (\tan x\cdot x^2) illustrates differentiating one factor at a time.

• Quotient rule: For (u(x)/v(x)), [ \left(\frac{u}{v}\right)’=\frac{u’v-uv’}{v^2}. ] The lesson applies this method to a quotient involving (\cot x) and (e^x).

3. Chain Rule

The chain rule is used to differentiate composite functions—functions with an inner function inside an outer function.

For (f(g(x))),

[ \frac{d}{dx}f(g(x))=f’(g(x))\,g’(x). ]

The procedure is:

1. Identify the outer function and the inner function.
2. Differentiate the outer function, keeping the inner expression in place.
3. Multiply by the derivative of the inner function.
4. For several layers of composition, continue applying the rule inward, one layer at a time.

Examples in the lesson involve trigonometric functions with expressions such as (5x) or (3x) inside them, exponential functions containing trigonometric expressions, and a product in which both factors are composite. The last type requires combining the product rule with the chain rule.

4. Choosing Which Rules to Apply

For the practice question involving a quotient with (\sin(8x)) in the numerator and (10x) in the denominator, the correct choices are:

• Use the quotient rule for the division.
• Use the chain rule when differentiating (\sin(8x)).

5. Opening Motivation Problem

The lesson asks whether (m^2+n^2) is always divisible by 4, even, odd, or a perfect square when (m) and (n) are odd integers. The answer is even.

Writing odd integers as (m=2a+1) and (n=2b+1),

[ \begin{aligned} m^2+n^2 &=(2a+1)^2+(2b+1)^2\ &=2(2a^2+2a+2b^2+2b+1), \end{aligned} ]

so the sum is twice an integer and is therefore even. The examples (3^2+5^2=34) and (1^2+5^2=26) also show that the result is not always divisible by 4 or odd.

6. Active Break

The instructor recites several poems as a short break between activities.

Speakers and Sources Featured

• Instructor/narrator: An unnamed presenter delivering the lesson for Grupo Educativo Monterrosales; no other named speaker is identified.
• Poem authors recited during the active break:
  • Federico García Lorca — “Caracola”
  • Blas de Otero — “Pato”
  • Rafael Alberti — “Nana de la tortuga”
  • Amado Nervo — “Mi gatito” (“My Kitten”)