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?

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!