Question

Y=x(4)/2+3x(3)-2x-12

Answer 1

`\displaystyle y' = -2x^3 + 9x^2 - 2`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Terms/Coefficients

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Explanation:

Step 1: Define

`\displaystyle y = -(x^4)/(2) + 3x^3 - 2x - 12`

Step 2: Differentiate

1. Basic Power Rule:
   `\displaystyle y' = -(4x^(4 - 1))/(2) + 3 \cdot 3x^(3 - 1) - 1 \cdot 2x^(1 - 1)`
2. Simplify Exponents:
   `\displaystyle y' = -(4x^3)/(2) + 3 \cdot 3x^2 - 1 \cdot 2`
3. Divide:
   `\displaystyle y' = -2x^3 + 3 \cdot 3x^2 - 1 \cdot 2`
4. Multiply:
   `\displaystyle y' = -2x^3 + 9x^2 - 2`