Question

How do you use the limit definition of the derivative to find the derivative of f(x)=1/x?

Answer 1

`\displaystyle f'(x) = (-1)/(x^2)`

General Formulas and Concepts:

Algebra I

Terms/Coefficients

• Expanding/Factoring

Functions

• Function Notation

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) (f(x + h) - f(x))/(h)`

Explanation:

Step 1: Define

Identify

`\displaystyle f(x) = (1)/(x)`

Step 2: Differentiate

1. Substitute in function [Definition of a Derivative]:
   `\displaystyle f'(x) = \lim_(h \to 0) ((1)/(x + h) - (1)/(x))/(h)`
2. Rewrite:
   `\displaystyle f'(x) = \lim_(h \to 0) (x - (x + h))/(hx(x + h))`
3. Expand:
   `\displaystyle f'(x) = \lim_(h \to 0) (x - x - h)/(hx(x + h))`
4. Combine like terms:
   `\displaystyle f'(x) = \lim_(h \to 0) (-h)/(hx(x + h))`
5. Simplify:
   `\displaystyle f'(x) = \lim_(h \to 0) (-1)/(x(x + h))`
6. Evaluate limit [Limit Rule - Variable Direct Substitution]:
   `\displaystyle f'(x) = (-1)/(x(x + 0))`
7. Simplify:
   `\displaystyle f'(x) = (-1)/(x^2)`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation