14. [-/2 Points] DETAILS MY NOTES HARMATHAP12 9.4.017. For the function \( P(x)=x^{3}-6 x \), at the point \( (2,-4) \), find the following. (a) the slope of the tangent to the curve (b) the instantaneous rate of change of the futtion Need Help? Readit SUBMIT ANSWER

Sure, let's solve each part step-by-step for the function \( P(x) = x^{3} - 6x \) at the point \( (2, -4) \). ### (a) Slope of the Tangent to the Curve To find the slope of the tangent to the curve at a specific point, we need to compute the derivative of the function and then evaluate it at the given \( x \)-value. 1. **Find the derivative \( P'(x) \):** \[ P(x) = x^{3} - 6x \] \[ P'(x) = \frac{d}{dx}(x^{3}) - \frac{d}{dx}(6x) = 3x^{2} - 6 \] 2. **Evaluate the derivative at \( x = 2 \):** \[ P'(2) = 3(2)^{2} - 6 = 3(4) - 6 = 12 - 6 = 6 \] **Therefore, the slope of the tangent to the curve at \( (2, -4) \) is **6**.** ### (b) Instantaneous Rate of Change of the Function The instantaneous rate of change of a function at a particular point is given by the value of its derivative at that point. Since we've already calculated \( P'(2) \) in part (a): \[ \text{Instantaneous Rate of Change at } x = 2 \text{ is } P'(2) = 6 \] **Therefore, the instantaneous rate of change of the function at \( (2, -4) \) is **6**.** --- **Final Answers:** - **(a)** The slope of the tangent to the curve at \( (2, -4) \) is **6**. - **(b)** The instantaneous rate of change of the function at \( (2, -4) \) is **6**.