Question

. Find the derivative of y = x² – 5x using the definition of the derivative.

Answer 1

`\displaystyle y' = 2x - 5`

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)`

Step-by-step explanation:

Step 1: Define

Identify

`\displaystyle y = x^2 - 5x`

Step 2: Differentiate

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

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

Unit: Differentiation