Answer:
[tex]\displaystyle f'(1) = 6[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
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
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle f(x) = 5x - 1 + \ln x[/tex]
Step 2: Differentiate
- [Function] Derivative Property [Addition/Subtraction]: [tex]\displaystyle f'(x) = \frac{d}{dx}[5x] - \frac{d}{dx}[1] + \frac{d}{dx}[\ln x][/tex]
- Rewrite [Derivative Property - Multiplied Constant]: [tex]\displaystyle f'(x) = 5 \frac{d}{dx}[x] - \frac{d}{dx}[1] + \frac{d}{dx}[\ln x][/tex]
- Derivative Rule [Basic Power Rule]: [tex]\displaystyle f'(x) = 5 - 0 + \frac{d}{dx}[\ln x][/tex]
- Logarithmic Differentiation: [tex]\displaystyle f'(x) = 5 + \frac{1}{x}[/tex]
- Substitute in x: [tex]\displaystyle f'(1) = 5 + \frac{1}{1}[/tex]
- Simplify: [tex]\displaystyle f'(1) = 6[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
