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resuuelve  y explica el paso a paso detalladamente trazando la tangente en la misma grafica

To solve the problem and estimate \( f'(3) \) using the graph of the function \( f(x) \), we'll follow these steps:

### Step 1: Understanding the Problem
You are asked to estimate the derivative \( f'(3) \), which represents the slope of the tangent line to the curve at \( x = 3 \).

### Step 2: Locate the Point on the Graph
Find the point on the graph where \( x = 3 \). According to the graph, the corresponding \( y \)-value (which is \( f(3) \)) is approximately \( 4.5 \).

### Step 3: Draw the Tangent Line
To estimate \( f'(3) \), draw a tangent line to the curve at the point where \( x = 3 \). This tangent line should just touch the curve at this point without crossing it. 

### Step 4: Estimate the Slope of the Tangent Line
The slope of the tangent line is calculated as the "rise over run," which is the change in \( y \) divided by the change in \( x \). 

- Identify two points on the tangent line: one at \( x = 3 \) and another point that lies on the tangent line to use for calculation.
- For instance, if the tangent line passes through another point such as \( (2, 2.5) \), you can use these points to find the slope.

\[ \text{Slope} = \frac{\Delta y}{\Delta x} = \frac{4.5 - 2.5}{3 - 2} = \frac{2}{1} = 2 \]

So, \( f'(3) \approx 2 \).

### Step 5: Conclude the Result
The estimated value of \( f'(3) \) is \( \boxed{2} \).

### Visualization
If you draw this on the graph, the tangent at \( x = 3 \) should look like a straight line just touching the curve at \( (3, 4.5) \) and the line should have a slope close to 2.

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Would you like more details on any of these steps or have any questions? 

Here are five related questions to deepen your understanding:

1. How do you find the derivative at a point for a non-linear function analytically?
2. What are some ways to accurately estimate a derivative if the graph isn't clear?
3. How does the slope of the tangent line relate to the behavior of the function at that point?
4. Can you estimate the derivative at other points on the graph using the same method?
5. How would the process differ if the function was given analytically rather than graphically?

**Tip:** When estimating derivatives from graphs, always try to choose points on the tangent line that are easy to read from the graph to minimize errors.

Learn how to estimate the derivative of a function at a specific point using tangent lines. This detailed guide covers finding the slope of a tangent line and applying it to estimate the derivative. Suitable for high school students studying calculus.

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