Answer:
[tex]\displaystyle y' = -100x^4(2x^5 - 1)[/tex]
General Formulas and Concepts:
Pre-Algebra
Algebra I
- Terms/Coefficients
- Factoring
- Functions
- Function Notation
Calculus
Derivatives
Derivative Notation
Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle y = (-4x^5 + 4)5x^5[/tex]
Step 2: Differentiate
- Product Rule: [tex]\displaystyle y' = \frac{d}{dx}[(-4x^5 + 4)]5x^5 + (-4x^5 + 4)\frac{d}{dx}[5x^5][/tex]
- Basic Power Rule [Derivative Property - Addition/Subtraction]: [tex]\displaystyle y' = (5 \cdot -4x^{5 - 1} + 0)5x^5 + (-4x^5 + 4)(5 \cdot 5x^{5 - 1})[/tex]
- Simplify: [tex]\displaystyle y' = (-20x^4)5x^5 + (-4x^5 + 4)(25x^4)[/tex]
- Factor: [tex]\displaystyle y' = 5x^4 \bigg[ (-20x^4)x + (-4x^5 + 4)5 \bigg][/tex]
- [Distributive Property] Distributive parenthesis: [tex]\displaystyle y' = 5x^4 \bigg[ -20x^5 - 20x^5 + 20 \bigg][/tex]
- Combine like terms: [tex]\displaystyle y' = 5x^4(-40x^5 + 20)[/tex]
- Factor: [tex]\displaystyle y' = -100x^4(2x^5 - 1)[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
