find the gradient of the curve whose equation is y=3xsquare at the point (2,12) step by step

To find the gradient of the curve \( y = 3x^2 \) at the point \( (2, 12) \), we'll follow these steps:

### Step 1: Differentiate the equation to find the derivative
The gradient of the curve at any point is given by the derivative of the function with respect to \( x \). So, we need to differentiate \( y = 3x^2 \) with respect to \( x \).

\[
\frac{dy}{dx} = \frac{d}{dx} (3x^2)
\]

Using the power rule for differentiation (\(\frac{d}{dx}(x^n) = nx^{n-1}\)):

\[
\frac{dy}{dx} = 3 \cdot 2x^{2-1} = 6x
\]

So, the derivative (gradient function) is:

\[
\frac{dy}{dx} = 6x
\]

### Step 2: Substitute the x-coordinate of the given point into the derivative
The point given is \( (2, 12) \), so \( x = 2 \). Substitute this into the derivative to find the gradient at this point.

\[
\text{Gradient at } (2, 12) = 6 \times 2 = 12
\]

### Conclusion
The gradient of the curve \( y = 3x^2 \) at the point \( (2, 12) \) is **12**.

Would you like more details or have any questions? Here are some related questions:

1. How do you find the gradient of a curve at different points?
2. What is the significance of the gradient at a specific point on a curve?
3. Can you explain the power rule for differentiation?
4. What happens to the gradient if the function is more complex, like \( y = 3x^3 \)?
5. How would you determine if the point \( (2, 12) \) lies on the curve without being told?
6. Can the gradient of a curve be negative? If yes, under what conditions?
7. How do you find the equation of the tangent line at a given point on a curve?
8. What are some applications of knowing the gradient of a curve in real-life scenarios?

**Tip:** Always check whether the given point lies on the curve by substituting the \( x \) value into the original equation before proceeding with further calculations.

Learn how to find the gradient of the curve y = 3x^2 at the point (2, 12) with a step-by-step solution. This problem involves differentiation, the power rule, and calculating the gradient at a specific point. Suitable for high school students (Grades 10-12).

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