Question

Please help! Calculus: Derivative by limit process!!

Answer 1

`\displaystyle f'(x) = 2x + 1`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Distributive Property

Algebra I

• Terms/Coefficients
• Expanding
• Factoring
• Functions
• Function Notation

Calculus

Limits

Limit Rule [Variable Direct Substitution]:
`\displaystyle \lim_(x \to c) x = c`

Derivatives

• 
  `\displaystyle f'(x) = \lim_(h \to 0) (f(x + h) - f(x))/(h)`

Derivative Notation

Explanation:

Step 1: Define

Identify

f(x) = x² + x - 2

Step 2: Differentiate

1. Substitute in function [Limit Process]:
   `\displaystyle f'(x) = \lim_(h \to 0) ([(x + h)^2 + (x + h) - 2] - (x^2 + x - 2))/(h)`
2. [Brackets] Expand:
   `\displaystyle f'(x) = \lim_(h \to 0) ([x^2 + 2hx + h^2 + x + h - 2] - (x^2 + x - 2))/(h)`
3. [Distributive Property] Distribute negative:
   `\displaystyle f'(x) = \lim_(h \to 0) (x^2 + 2hx + h^2 + x + h - 2 - x^2 - x + 2)/(h)`
4. Combine like terms:
   `\displaystyle f'(x) = \lim_(h \to 0) (2hx + h^2 + h)/(h)`
5. Factor:
   `\displaystyle f'(x) = \lim_(h \to 0) (h(2x + h + 1))/(h)`
6. Simplify:
   `\displaystyle f'(x) = \lim_(h \to 0) 2x + h + 1`
7. Evaluate limit [Limit Rule - Variable Direct Substitution]:
   `\displaystyle f'(x) = 2x + 0 + 1`
8. Simplify:
   `\displaystyle f'(x) = 2x + 1`

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

Unit: Derivatives

Book: College Calculus 10e