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AysviL [449]
3 years ago
6

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

Engineering
1 answer:
gavmur [86]3 years ago
8 0

Answer:

\displaystyle y' = 2x - 5

General Formulas and Concepts:

<u>Algebra I</u>

Terms/Coefficients

  • Expanding/Factoring

Functions

  • Function Notation

<u>Calculus</u>

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} \frac{f(x + h) - f(x)}{h}

Explanation:

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle y = x^2 - 5x

<u>Step 2: Differentiate</u>

  1. Substitute in function [Definition of a Derivative]:                                       \displaystyle y' = \lim_{h \to 0} \frac{[(x + h)^2 - 5(x + h)] - (x^2 - 5x)}{h}
  2. Expand:                                                                                                         \displaystyle y' = \lim_{h \to 0} \frac{x^2 + 2hx + h^2 - 5x - 5h - x^2 + 5x}{h}
  3. Combine like terms:                                                                                     \displaystyle y' = \lim_{h \to 0} \frac{2hx + h^2 - 5h}{h}
  4. Factor:                                                                                                           \displaystyle y' = \lim_{h \to 0} \frac{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

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