Answer:
\(\displaystyle y''' = 2 \sec^2 (x) \bigg( 2\tan^2 (x) + \sec^2 (x) \bigg)\)
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: \(\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)\)
Derivative Property [Addition/Subtraction]: \(\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]\)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]: \(\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)\)
Derivative Rule [Chain Rule]: \(\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)\)
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle y = \tan x\)
Step 2: Differentiate
- Trigonometric Differentiation: \(\displaystyle y' = \sec^2 (x)\)
- Basic Power Rule [Derivative Rule - Chain Rule]: \(\displaystyle y'' = 2 \sec (x) \cdot \frac{d}{dx}[\sec (x)]\)
- Trigonometric Differentiation: \(\displaystyle y'' = 2 \sec (x) \cdot \sec (x) \tan (x)\)
- Simplify: \(\displaystyle y'' = 2 \sec^2 (x) \tan (x)\)
- Derivative Rule [Product Rule]: \(\displaystyle y''' = \frac{d}{dx}[2 \sec^2 (x)] \tan (x) + 2 \sec^2 (x) \frac{d}{dx}[\tan (x)]\)
- Rewrite [Derivative Property - Multiplied Constant]: \(\displaystyle y''' = 2 \frac{d}{dx}[\sec^2 (x)] \tan (x) + 2 \sec^2 (x) \frac{d}{dx}[\tan (x)]\)
- Trigonometric Differentiation: \(\displaystyle y''' = 2 \frac{d}{dx}[\sec^2 (x)] \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)\)
- Basic Power Rule [Derivative Rule - Chain Rule]: \(\displaystyle y''' = 2 \big( 2 \sec (x) \big) \frac{d}{dx}[\sec (x)] \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)\)
- Trigonometric Differentiation: \(\displaystyle y''' = 2 \big( 2 \sec (x) \big) \big( \sec (x) \tan (x) \big) \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)\)
- Simplify: \(\displaystyle y''' = 4 \sec^2 (x) \tan^2 (x) + 2 \sec^4 (x)\)
- Factor: \(\displaystyle y''' = 2 \sec^2 (x) \bigg( 2\tan^2 (x) + \sec^2 (x) \bigg)\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
