https://file.aiquickdraw.com/mathbot/file/1730979755862hgnnnh4j.jpg

The task here is to find the derivative \( f'(x_0) \) at the point \( x_0 \) using the graph provided. This derivative corresponds to the slope of the tangent line drawn at \( x = x_0 \) on the curve \( y = f(x) \).

Here’s how to approach it:

1. **Identify the Point**: From the graph, \( x_0 \) is marked at \( x = -3 \).
2. **Determine the Tangent’s Slope**: The slope of the tangent line at this point is the derivative \( f'(x_0) \).
   - The tangent line is shown in blue and appears to have a negative slope.
3. **Calculate the Slope**:
   - To determine the slope, pick two points on the tangent line. For example:
     - Point 1: Approximately at \( (-5, -1) \)
     - Point 2: Approximately at \( (-3, -2) \)
   - Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
     \[
     m = \frac{-2 - (-1)}{-3 - (-5)} = \frac{-1}{2} = -0.5
     \]
   - Therefore, \( f'(x_0) = -0.5 \).

### Answer:
The value of \( f'(x_0) \) in decimal form is:
\[
\text{Atsakymas: } -0.5
\]

Would you like further details on interpreting graphs for derivatives?

### Further Related Questions
1. How do you find the slope of a line between two points?
2. What does the slope of a tangent line represent in calculus?
3. How can you approximate slopes if points aren't clearly marked?
4. How does a derivative relate to the function’s rate of change?
5. What are the steps to find the derivative at a specific point on any graph?

### Tip
When interpreting tangent lines on graphs, always check for clear points to estimate the slope accurately.

Nubraižytas funkcijos y = f(x) grafikas ir jo liestinė, einanti per tašką x0. Apskaičiuok f'(x0) reikšmę. Užrašyk ją dešimtaine trupmena.

Learn how to calculate the derivative f'(x0) at a specific point x0 on the function y = f(x) by finding the slope of the tangent line passing through this point. This problem involves calculus concepts such as derivatives and the slope formula, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation