Question

Find the derivative of \( f(x) = 4x^3 - 6x^2 + 2x - 5 \).

• A. \( 12x^2 - 12x + 2 \)
• B. \( 12x^2 - 10x + 2 \)
• C. \( 12x^2 - 12x + 5 \)
• D. \( 12x^2 - 10x + 3 \)

Hint:

Remember: Use the power rule for differentiation for each term of the polynomial function.

Answer 1

The Correct Option is A

Step 1: Use the power rule for differentiation
The power rule for differentiation states that the derivative of \( ax^n \) is \( a \cdot n \cdot x^{n-1} \).
Step 2: Differentiate each term We are given: \[ f(x) = 4x^3 - 6x^2 + 2x - 5 \] Now, differentiate each term:
- The derivative of \( 4x^3 \) is \( 12x^2 \),
- The derivative of \( -6x^2 \) is \( -12x \),
- The derivative of \( 2x \) is \( 2 \),
- The derivative of the constant \( -5 \) is \( 0 \).
Thus, the derivative of \( f(x) \) is: \[ f'(x) = 12x^2 - 12x + 2 \] Answer: Therefore, the derivative of \( f(x) = 4x^3 - 6x^2 + 2x - 5 \) is \( 12x^2 - 12x + 2 \). So, the correct answer is option (1).