Respuesta :

Space

Answer:

f'(x) = 6x² + 12x + 5

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

Distributive Property

Algebra I

  • Terms/Coefficients/Degrees

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Product Rule: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]

Step-by-step explanation:

Step 1: Define

f(x) = (x + 3)(2x² + 5)

Step 2: Differentiate

  1. Product Rule [Basic Power Rule]:                                                                 f'(x) = (1 · x¹⁻¹ + 0)(2x² + 5) + (x + 3)(2 · 2x²⁻¹ + 0)
  2. [Derivative] Simplify:                                                                                       f'(x) = (1)(2x² + 5) + (x + 3)(4x)
  3. [Derivative] Multiply:                                                                                       f'(x) = 2x² + 5 + (x + 3)(4x)
  4. [Derivative] Distribute:                                                                                     f'(x) = 2x² + 5 + 4x² + 12x
  5. [Derivative] Combine like terms:                                                                   f'(x) = 6x² + 12x + 5

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