To find the slope of the tangent to the curve given by the equation \( y = 3x^2 - 2x + 5 \) at the point \( (1, 6) \), we need to follow these steps: ### Step 1: Differentiate the Function The first step is to find the derivative of the function \( y \) with respect to \( x \). The derivative, denoted as \( \frac{dy}{dx} \), gives us the slope of the tangent line at any point on the curve. The function is: \[ y = 3x^2 - 2x + 5 \] To differentiate, we apply the power rule: - The derivative of \( 3x^2 \) is \( 6x \) (using \( \frac{d}{dx}(x^n) = nx^{n-1} \)). - The derivative of \( -2x \) is \( -2 \). - The derivative of the constant \( 5 \) is \( 0 \). Putting it all together, we have: \[ \frac{dy}{dx} = 6x - 2 \] ### Step 2: Evaluate the Derivative at the Given Point Next, we need to evaluate the derivative at the specific point \( x = 1 \) to find the slope of the tangent line at that point. Substituting \( x = 1 \) into the derivative: \[ \frac{dy}{dx} \bigg|_{x=1} = 6(1) - 2 = 6 - 2 = 4 \] ### Conclusion Thus, the slope of the tangent to the curve at the point \( (1, 6) \) is \( 4 \). ### Answer The correct option is **A. 4**. ### Explanation of Other Options - **B. 1**: This option is incorrect because it does not reflect the calculated slope from the derivative. The slope at \( x = 1 \) is \( 4 \), not \( 1 \). - **C. 6**: This option is also incorrect. While \( 6 \) is the coefficient of \( x^2 \) in the original function, it does not represent the slope at the point \( (1, 6) \). - **D. 5**: This option is incorrect as well. The value \( 5 \) is the constant term in the function, which does not relate to the slope of the tangent line. ### Common Pitfalls - **Misunderstanding the Derivative**: Some students may confuse the function value at a point with the slope. Remember, the derivative gives the slope, not the function's output. - **Forgetting to Substitute**: After finding the derivative, it’s crucial to substitute the correct \( x \) value to find the slope at that specific point. ### Revision Summary - The slope of the tangent line is found using the derivative of the function. - Differentiate \( y = 3x^2 - 2x + 5 \) to get \( \frac{dy}{dx} = 6x - 2 \). - Evaluate the derivative at the point of interest, \( x = 1 \), to find the slope. - The correct slope at \( (1, 6) \) is \( 4 \), corresponding to option A.