Question

F(x) = (x+3)(2x²+5) Find the derivative

Answer 1

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: $\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

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