Question

Find the gradient of the curve y=9-3^2 at a point where x=-2. Pls explain in a simple way that i can understand.

Answer 1

To find the gradient of the curve given by the equation \( y = 9 - 3x^2 \) at the point where \( x = -2 \), we need to follow these steps:

1. **Understand the function**: Here, we have a function of \( x \), which means we are dealing with a curve. The gradient (or slope) of the curve at any point can be found by calculating the derivative of the function.

2. **Differentiate the function**: The gradient is found by calculating the derivative of \( y \) with respect to \( x \).

Starting with our function:
\[
y = 9 - 3x^2
\]

We differentiate this with respect to \( x \):
\[
\frac{dy}{dx} = 0 - 6x = -6x
\]

3. **Evaluate the derivative at \( x = -2 \)**: We now substitute \( x = -2 \) into the derivative we found:
\[
\frac{dy}{dx} = -6(-2) = 12
\]

4. **Interpret the result**: The value \( 12 \) represents the gradient of the curve \( y = 9 - 3x^2 \) at the point where \( x = -2 \). This means that at \( x = -2 \), the slope of the curve is 12, indicating that the curve is rising steeply at that point.

### Summary:
- The gradient of the curve \( y = 9 - 3x^2 \) at \( x = -2 \) is **12**.