Question

Help differentiate this

Answer 1

Answer:

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

General Formulas and Concepts:

Pre-Algebra

• Distributive Property

Algebra I

• Terms/Coefficients
• Expand by FOIL
• Functions
• Function Notation

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:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹  

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

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

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Derivatives

Book: College Calculus 10e