Question

How do you find the derivative of y=√x using the definition of derivative?

Answer 1

`\displaystyle y' = (1)/(2√(x))`

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 y = √(x)`

Step 2: Differentiate

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

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

Unit: Differentiation