Question

Please help! Calculus: Derivative by limit process!!

Answer 1

Answer:

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

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

• $\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Derivative Notation

Step-by-step explanation:

Step 1: Define

Identify

f(x) = x² + x - 2

Step 2: Differentiate

1. Substitute in function [Limit Process]:                                                             $\displaystyle f'(x) = \lim_{h \to 0} \frac{[(x + h)^2 + (x + h) - 2] - (x^2 + x - 2)}{h}$
2. [Brackets] Expand:                                                                                            $\displaystyle f'(x) = \lim_{h \to 0} \frac{[x^2 + 2hx + h^2 + x + h - 2] - (x^2 + x - 2)}{h}$
3. [Distributive Property] Distribute negative:                                                    $\displaystyle f'(x) = \lim_{h \to 0} \frac{x^2 + 2hx + h^2 + x + h - 2 - x^2 - x + 2}{h}$
4. Combine like terms:                                                                                         $\displaystyle f'(x) = \lim_{h \to 0} \frac{2hx + h^2 + h}{h}$
5. Factor:                                                                                                               $\displaystyle f'(x) = \lim_{h \to 0} \frac{h(2x + h + 1)}{h}$
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