Question

Using first principle find derivatives of x2
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Answer 1

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

General Formulas and Concepts:

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) = x^2`

Step 2: Differentiate

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

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

Unit: Differentiation