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

Sagot :

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Answer:

[tex]\displaystyle y' = 2x - 5[/tex]

General Formulas and Concepts:

Algebra I

Terms/Coefficients

  • Expanding/Factoring

Functions

  • Function Notation

Calculus

Limits

Limit Rule [Variable Direct Substitution]:                                                             [tex]\displaystyle \lim_{x \to c} x = c[/tex]

Differentiation

  • Derivatives
  • Derivative Notation
  • Definition of a Derivative:                                                                             [tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]

Explanation:

Step 1: Define

Identify

[tex]\displaystyle y = x^2 - 5x[/tex]

Step 2: Differentiate

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

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

Unit: Differentiation

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