This is a specific request for a study guide based on a well-known textbook in the Philippines and other Southeast Asian countries: "Differential and Integral Calculus" by Feliciano and Uy . Note on Edition: Most standard editions of Feliciano & Uy cover Chapter 4: Applications of Trigonometric Functions (or sometimes Transcendental Functions ). However, some older editions place Applications of Derivatives in Chapter 4. Given the progression of calculus, Chapter 4 most commonly deals with Derivatives of Trigonometric Functions and their basic applications. I will provide a guide based on the most likely content of Chapter 4 : Derivatives of Trigonometric Functions and the Chain Rule applied to them.
Study Guide: Chapter 4 of Feliciano & Uy Topic: Differentiation of Trigonometric Functions I. Core Objective of the Chapter By the end of this chapter, you should be able to derive trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) and solve basic application problems (slopes, rates of change, velocity). II. Must-Memorize Derivative Formulas (The "Holy 6") Let ( u ) be a differentiable function of ( x ).
Sine: ( \frac{d}{dx}(\sin u) = \cos u \cdot \frac{du}{dx} ) Cosine: ( \frac{d}{dx}(\cos u) = -\sin u \cdot \frac{du}{dx} ) Tangent: ( \frac{d}{dx}(\tan u) = \sec^2 u \cdot \frac{du}{dx} ) Cotangent: ( \frac{d}{dx}(\cot u) = -\csc^2 u \cdot \frac{du}{dx} ) Secant: ( \frac{d}{dx}(\sec u) = \sec u \tan u \cdot \frac{du}{dx} ) Cosecant: ( \frac{d}{dx}(\csc u) = -\csc u \cot u \cdot \frac{du}{dx} )
Feliciano & Uy Tip: They emphasize the negative signs for cosine, cotangent, and cosecant. Do not forget them on exams. This is a specific request for a study
III. Step-by-Step Workflow for Solving Problems Step 1: Identify the outer trigonometric function (sin, cos, tan, etc.). Step 2: Identify ( u ) (the inside function). Step 3: Differentiate the outer function (keeping ( u ) intact). Step 4: Multiply by ( \frac{du}{dx} ) (derivative of the inside). Step 5: Simplify using algebraic identities (e.g., ( \sin^2 x + \cos^2 x = 1 )). IV. Common Problem Types in Feliciano & Uy (with Examples) Type A: Direct Differentiation (No Chain Rule needed) Example: ( y = \sin x + \cos x )
Solution: ( y' = \cos x - \sin x )
Type B: Chain Rule (Most problems in this chapter) Example: ( y = \sin(5x^2) ) Given the progression of calculus, Chapter 4 most
Identify ( u ): ( u = 5x^2 ) ( \frac{dy}{du} ): ( \cos u ) ( \frac{du}{dx} ): ( 10x ) Answer: ( y' = \cos(5x^2) \cdot 10x )
Type C: Product/Quotient Rule with Trig Example: ( y = x^2 \tan x )
Use Product Rule: ( y' = (2x)(\tan x) + (x^2)(\sec^2 x) ) Final: ( y' = 2x \tan x + x^2 \sec^2 x ) Core Objective of the Chapter By the end
Type D: Trigonometric Identities Required (Classic Feliciano & Uy style) Example: ( y = \frac{\sin x}{1 + \cos x} )
Before differentiating , simplify using identity: