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 $$ (2,-4) $$ Entr an exact number (b) the instantaneous rate of change of the function Need Help? Rasast

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Let's analyze the function $$ P(x) = x^{3} - 6x $$ at the point $$ (2, -4) $$.

### Part (a): Slope of the Tangent to the Curve at $$ (2, -4) $$

To find the slope of the tangent line at a specific point, we need to compute the derivative of $$ P(x) $$ and evaluate it at $$ x = 2 $$.

1. **Find the derivative $$ P'(x) $$:**
   $$
   P'(x) = \frac{d}{dx}(x^{3} - 6x) = 3x^{2} - 6
   $$

2. **Evaluate the derivative at $$ x = 2 $$:**
   $$
   P'(2) = 3(2)^{2} - 6 = 3 \times 4 - 6 = 12 - 6 = 6
   $$

**Therefore, the slope of the tangent at $$ (2, -4) $$ is** $$ \boxed{6} $$.

### Part (b): Instantaneous Rate of Change of the Function at $$ (2, -4) $$

The **instantaneous rate of change** of a function at a specific point is essentially the slope of the tangent line at that point. From part (a), we've already determined that:

$$
\text{Instantaneous Rate of Change at } x = 2 \text{ is } 6
$$

**So, the instantaneous rate of change of the function at $$ (2, -4) $$ is** $$ \boxed{6} $$.
The slope of the tangent at $$ (2, -4) $$ is 6, and the instantaneous rate of change of the function at that point is also 6.

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 \( (2,-4) \) Entr an exact number (b) the instantaneous rate of change of the function Need Help? Rasast

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