Question

Product and quotient rule ,differentiate each function with respect to x, in calculus

Answer 1

Answer:

$\displaystyle y' = -100x^4(2x^5 - 1)$

General Formulas and Concepts:

Pre-Algebra

• Distributive Property

Algebra I

• Terms/Coefficients
• Factoring
• 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

$\displaystyle y = (-4x^5 + 4)5x^5$

Step 2: Differentiate

1. Product Rule:                                                                                                    $\displaystyle y' = \frac{d}{dx}[(-4x^5 + 4)]5x^5 + (-4x^5 + 4)\frac{d}{dx}[5x^5]$
2. Basic Power Rule [Derivative Property - Addition/Subtraction]:                   $\displaystyle y' = (5 \cdot -4x^{5 - 1} + 0)5x^5 + (-4x^5 + 4)(5 \cdot 5x^{5 - 1})$
3. Simplify:                                                                                                             $\displaystyle y' = (-20x^4)5x^5 + (-4x^5 + 4)(25x^4)$
4. Factor:                                                                                                               $\displaystyle y' = 5x^4 \bigg[ (-20x^4)x + (-4x^5 + 4)5 \bigg]$
5. [Distributive Property] Distributive parenthesis:                                            $\displaystyle y' = 5x^4 \bigg[ -20x^5 - 20x^5 + 20 \bigg]$
6. Combine like terms:                                                                                         $\displaystyle y' = 5x^4(-40x^5 + 20)$
7. Factor:                                                                                                               $\displaystyle y' = -100x^4(2x^5 - 1)$

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

Unit: Derivatives

Book: College Calculus 10e