Answer:
[tex]\displaystyle \frac{d}{dx}[\sec x] = \sec x \tan x[/tex]
General Formulas and Concepts:
Pre-Calculus
Calculus
Differentiation
- Derivatives
- Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Quotient Rule]: [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Step-by-step explanation:
*Note:
This is a known trigonometric derivative.
Step 1: Define
Identify
[tex]\displaystyle y = \sec x[/tex]
Step 2: Differentiate
- Rewrite [Trigonometric Identities]: [tex]\displaystyle y = \frac{1}{\cos x}[/tex]
- Derivative Rule [Quotient Rule]: [tex]\displaystyle y' = \frac{(1)' \cos x - 1(\cos x)'}{\cos^2 x}[/tex]
- Basic Power Rule: [tex]\displaystyle y' = \frac{(0) \cos x - 1(\cos x)'}{\cos^2 x}[/tex]
- Trigonometric Differentiation: [tex]\displaystyle y' = \frac{(0) \cos x + 1(\sin x)}{\cos^2 x}[/tex]
- Simplify: [tex]\displaystyle y' = \frac{\sin x}{\cos^2 x}[/tex]
- Rewrite: [tex]\displaystyle y' = \frac{\sin x}{\cos x} \cdot \frac{1}{\cos x}[/tex]
- Rewrite [Trigonometric Identities]: [tex]\displaystyle y' = \tan x \sec x[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
