https://file.aiquickdraw.com/mathbot/file/1727347758170hi3vfhcc.jpg

The problem asks to find the derivative of the function \( f(x) \) at the point \( x_0 \), and the image shows a graph of the function \( y = f(x) \) and its tangent line at \( x_0 \).

### Key Observations:
1. **Derivative Interpretation**: The derivative of a function at a point \( x_0 \), denoted as \( f'(x_0) \), is the slope of the tangent line to the graph of the function at that point.
2. **Tangent Line Slope**: From the graph, we can observe the green tangent line, and we need to determine its slope.

#### Determining the slope:
- The graph suggests the following:
  - At \( x = 0 \), the tangent line has a height of \( y = 1 \).
  - At \( x = 1 \), the tangent line has a height of \( y = 2 \).
  
Thus, over the interval from \( x = 0 \) to \( x = 1 \), the tangent line rises by 1 unit (from 1 to 2) and runs by 1 unit (from \( x = 0 \) to \( x = 1 \)).

Therefore, the slope of the tangent line is:
\[
f'(x_0) = \frac{\Delta y}{\Delta x} = \frac{2 - 1}{1 - 0} = 1
\]

### Conclusion:
The value of the derivative \( f'(x_0) \) is \( \boxed{1} \).

Would you like further details or explanations on any step? Here are some related questions for you to consider:

1. How do you compute the derivative of a function algebraically?
2. What are the key characteristics of a tangent line to a curve?
3. What does the slope of a tangent line tell us about the behavior of a function?
4. How would the graph change if the derivative were negative?
5. What is the significance of the point where the tangent line meets the curve?

**Tip**: When interpreting graphs, always check the rise and run to find the slope accurately, especially when dealing with tangent lines.

The figure shows the graph of the function y = f(x) and the tangent to it at the point with the abscissa x₀. Find the value of the derivative of the function f(x) at the point x₀.

This problem involves finding the derivative of a function at the point x₀ using the graph of the function and its tangent line. The derivative is calculated as the slope of the tangent line. Suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation