To find the derivative of a function roughly on a graph, follow these steps:

### Steps to Estimate Derivative on a Graph:
1. **Identify a Point on the Graph:**
   Choose a point \( P(x_0, y_0) \) on the curve where you want to find the derivative.

2. **Draw the Tangent Line:**
   Draw a tangent line to the graph at that point. A tangent line touches the curve at exactly one point and represents the instantaneous rate of change at that point.

3. **Estimate the Slope of the Tangent Line:**
   - The derivative is the slope of this tangent line.
   - To estimate the slope, select two points on the tangent line and use the slope formula:
     \[
     \text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
     \]
   - \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two points on the tangent line.

4. **Interpret the Result:**
   The calculated slope gives you an approximation of the derivative at \( x_0 \), or \( f'(x_0) \).

### Example (without graph):
- If the tangent line passes through points \( (1, 2) \) and \( (3, 6) \), the slope (derivative) would be:
  \[
  \text{slope} = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2
  \]

Thus, the derivative at the point \( x_0 = 1 \) is roughly 2.

Would you like more details on any part of the process or need further clarification?

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Here are 5 follow-up questions to consider:
1. How can we identify points of zero derivative on a graph?
2. What happens when the graph has a vertical tangent line?
3. How does the second derivative relate to the shape of the graph?
4. How would you estimate the derivative at a point where the graph changes concavity?
5. What is the significance of a positive or negative derivative on a graph?

**Tip:** The steeper the tangent line, the larger the magnitude of the derivative!

Learn how to estimate the derivative of a function on a graph by drawing a tangent line and calculating its slope. This step-by-step guide explains key concepts of derivatives and slope. Ideal for high school students in Grades 11-12.

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