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Answer:

$\displaystyle y' = 20x^3 + 6x^2 + 70x + 9$

General Formulas and Concepts:

Pre-Algebra

Algebra I

Calculus

Derivatives

Derivative Notation

Derivative Property [Multiplied Constant]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]: $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

Derivative Rule [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

Identify

$\displaystyle y = (x^3 + 7x - 1)(5x + 2)$

Step 2: Differentiate

Product Rule: $\displaystyle y' = \frac{d}{dx}[(x^3 + 7x - 1)](5x + 2) + (x^3 + 7x - 1)\frac{d}{dx}[(5x + 2)]$
Basic Power Rule [Derivative Property - Addition/Subtraction]: $\displaystyle y' = (3x^{3 - 1}+ 7x^{1 - 1} - 0)(5x + 2) + (x^3 + 7x - 1)(5x^{1 - 1} + 0)$
Simplify: $\displaystyle y' = (3x^2+ 7)(5x + 2) + 5(x^3 + 7x - 1)$
Expand: $\displaystyle y' = 15x^3 + 6x^2 + 35x + 14 + 5(x^3 + 7x - 1)$
[Distributive Property] Distribute 5: $\displaystyle y' = 15x^3 + 6x^2 + 35x + 14 + 5x^3 + 35x - 5$
Combine like terms: $\displaystyle y' = 20x^3 + 6x^2 + 70x + 9$