Answer:
[tex]\displaystyle f'(x) = \frac{-1}{x^2}[/tex]
General Formulas and Concepts:
Algebra I
Terms/Coefficients
Functions
Calculus
Limits
Limit Rule [Variable Direct Substitution]: [tex]\displaystyle \lim_{x \to c} x = c[/tex]
Differentiation
- Derivatives
- Derivative Notation
- Definition of a Derivative: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle f(x) = \frac{1}{x}[/tex]
Step 2: Differentiate
- Substitute in function [Definition of a Derivative]: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h}[/tex]
- Rewrite: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{x - (x + h)}{hx(x + h)}[/tex]
- Expand: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{x - x - h}{hx(x + h)}[/tex]
- Combine like terms: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{-h}{hx(x + h)}[/tex]
- Simplify: [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{-1}{x(x + h)}[/tex]
- Evaluate limit [Limit Rule - Variable Direct Substitution]: [tex]\displaystyle f'(x) = \frac{-1}{x(x + 0)}[/tex]
- Simplify: [tex]\displaystyle f'(x) = \frac{-1}{x^2}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
