Answer:
E. [tex]\displaystyle f'(s) = s^5 \sec^2 s + 5s^4 \tan x[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle f(s) = s^5 \tan s[/tex]
Step 2: Differentiate
- Derivative Rule [Product Rule]: [tex]\displaystyle f'(s) = (s^5)' \tan s + s^5(\tan s)'[/tex]
- Basic Power Rule/Trigonometric Differentiation: [tex]\displaystyle f'(s) = 5s^4 \tan s + s^5 \sec^2 x[/tex]
- Rewrite: [tex]\displaystyle f'(s) = s^5 \sec^2 s + 5s^4 \tan x[/tex]
∴ Our answer is E.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
