Question

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

Answer 1

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 (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

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