Question

Find the derivative of the function \( f(x) = 3x^2 - 5x + 7 \).

• A. \( 6x - 5 \)
• B. \( 6x + 5 \)
• C. \( 3x^2 + 5 \)
• D. \( 3x^2 - 5 \)

Hint:

Remember: Use the power rule for differentiating polynomials. The derivative of a constant is zero, and the derivative of \( ax^n \) is \( n \cdot ax^{n-1} \).

Answer 1

The Correct Option is A

Step 1: Apply the power rule for differentiation The power rule dictates that for a function \( f(x) = ax^n \), its derivative is \( f'(x) = n \cdot ax^{n-1} \).Step 2: Differentiate each term of the given function The function is \( f(x) = 3x^2 - 5x + 7 \). We differentiate each term individually:1. Differentiating \( 3x^2 \) yields: \[ \frac{d}{dx}(3x^2) = 6x \]2. Differentiating \( -5x \) results in: \[ \frac{d}{dx}(-5x) = -5 \]3. The derivative of the constant \( 7 \) is: \[ \frac{d}{dx}(7) = 0 \]Step 3: Sum the derivatives of the terms Combining the derivatives of \( f(x) = 3x^2 - 5x + 7 \) yields:\[f'(x) = 6x - 5\]Answer: Consequently, the derivative of the function is \( 6x - 5 \). This corresponds to option (1).