Answer :
Answer:
[tex]\displaystyle \frac{dy}{dx} = 2x(10x^3 - 30x - 7)[/tex]
General Formulas and Concepts:
Algebra I
- Terms/Coefficients
- Factoring/Expanding
Calculus
Differentiation
- 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
Explanation:
Step 1: Define
Identify
[tex]\displaystyle 4x^2(x^3 - 5x) - 7x^2[/tex]
Step 2: Differentiate
- Expand: [tex]\displaystyle 4x^5 - 20x^3 - 7x^2[/tex]
- Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{dy}{dx} = \frac{d}{dx}[4x^5] - \frac{d}{dx}[20x^3] - \frac{d}{dx}[7x^2][/tex]
- Derivative Property [Multiplied Constant]: [tex]\displaystyle \frac{dy}{dx} = 4\frac{d}{dx}[x^5] - 20\frac{d}{dx}[x^3] - 7\frac{d}{dx}[x^2][/tex]
- Basic Power Rule: [tex]\displaystyle \frac{dy}{dx} = 4(5x^4) - 20(3x^2) - 7(2x)[/tex]
- Simplify: [tex]\displaystyle \frac{dy}{dx} = 20x^4 - 60x^2 - 14x[/tex]
- Factor: [tex]\displaystyle \frac{dy}{dx} = 2x(10x^3 - 30x - 7)[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Book: College Calculus 10e