Question

PLEASE HELP!!! CALCULUS ASSIGNMENT

Answer 1

( d ) 12x³ - 15x² + 2

Explanation:

y = 3x⁴ - 5x³ + 2x - 1

• The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of axⁿ is nax^{n-1}.

`\small \sf \: y = 4* 3x^(4-1)+3\left(-5\right)x^(3-1)+2x^(1-1)`

• multiply 4 × 3.

`\small \sf \: y = 12x^(4-1)+3\left(-5\right)x^(3-1)+2x^(1-1)`

• Multiply 3 × -5

`\small \sf \: y = 12x^(3)-15x^(3-1)+2x^(1-1)`

• subtract the exponents

y = 12x³ - 15x² + 2⁰

• For any term t except 0, t⁰ = 1.

y = 12x³ - 15x² + 2 × 1

y = 12x³ - 15x² + 2

Hence, option ( d ) is the correct answer.

Answer 2

(d)
`\displaystyle 12x^3 - 15x^2 + 2`

General Formulas and Concepts:

Algebra I

• Functions
• Function Notation

Calculus

Derivatives

Derivative Notation

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Explanation:

Step 1: Define

Identify

`\displaystyle y = 3x^4 - 5x^3 + 2x - 1`

Step 2: Differentiate

1. Basic Power Rule:
   `\displaystyle y' = 4(3x^(4 - 1)) - 3(5x^(3 - 1)) + 2x^(1 - 1) - 0`
2. Simplify:
   `\displaystyle y' = 4(3x^3) - 3(5x^2) + 2`
3. Multiply:
   `\displaystyle y' = 12x^3 - 15x^2 + 2`

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

Unit: Derivatives

Book: College Calculus 10e