Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation3h 18m
- 4. Applications of Derivatives2h 38m
- 5. Graphical Applications of Derivatives6h 2m
- 6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
- 7. Antiderivatives & Indefinite Integrals1h 26m
3. Techniques of Differentiation
Derivatives of Trig Functions
4:35 minutes
Problem 3.5.53
Textbook Question
Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (sec x) = sec x tan x
Verified step by step guidance
1
Step 1: Recall the definition of secant. The secant function is defined as \( \sec x = \frac{1}{\cos x} \).
Step 2: Apply the Quotient Rule. The Quotient Rule states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u = 1 \) and \( v = \cos x \).
Step 3: Differentiate \( u \) and \( v \). Since \( u = 1 \), \( u' = 0 \). For \( v = \cos x \), \( v' = -\sin x \).
Step 4: Substitute into the Quotient Rule formula. Plug \( u, v, u', \) and \( v' \) into the formula: \( \frac{\cos x \cdot 0 - 1 \cdot (-\sin x)}{(\cos x)^2} \).
Step 5: Simplify the expression. The expression simplifies to \( \frac{\sin x}{\cos^2 x} \), which can be rewritten as \( \sec x \tan x \) using trigonometric identities.
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