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