Step 1: Apply the power rule of 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 individually Given the function:\[f(x) = 4x^3 - 6x^2 + 2x - 5\]Differentiating each term yields:- Derivative of \( 4x^3 \): \( 12x^2 \),
- Derivative of \( -6x^2 \): \( -12x \),
- Derivative of \( 2x \): \( 2 \),
- Derivative of the constant \( -5 \): \( 0 \).
Combining these results, the derivative of \( f(x) \) is:\[f'(x) = 12x^2 - 12x + 2\]Answer: The derivative of \( f(x) = 4x^3 - 6x^2 + 2x - 5 \) is \( 12x^2 - 12x + 2 \). This corresponds to option (1).