https://r2.erweima.ai/mathbot/image/1725841335602ctzf5fyx.jpg

It appears that you've uploaded an image. Let me extract and analyze the mathematical content from the image.

One moment while I process it.The image you provided contains a mathematical graph and a caption related to a function.

- The caption says: "Find \( f'(1) \), where \( f(x) = x^2 - 2x \)."
- There is a graphical representation, showing a function \( f(x) = x^2 - 2x \) with a tangent line at \( x = 1 \), and the goal is to find the derivative \( f'(1) \).

### Steps to Solve:
1. The function is \( f(x) = x^2 - 2x \).
2. To find \( f'(x) \), we first compute the derivative of the function.

   \[
   f'(x) = \frac{d}{dx} (x^2 - 2x) = 2x - 2
   \]

3. Now, substitute \( x = 1 \) into the derivative.

   \[
   f'(1) = 2(1) - 2 = 0
   \]

Thus, \( f'(1) = 0 \).

This means the slope of the tangent line to the curve at \( x = 1 \) is 0, which is why the graph likely shows a horizontal tangent at that point.

Would you like more details on the graph interpretation or have any specific questions?

### 5 Questions to Explore:
1. What does the derivative tell us about the behavior of the function at a given point?
2. How do you compute the tangent line at a specific point using derivatives?
3. What is the significance of finding where the derivative equals zero?
4. How do we apply the power rule in derivatives?
5. How can graphing help verify the results of calculus computations?

### Tip:
To visually confirm your derivative result, observe that when the slope \( f'(x) \) is zero, the tangent line at that point should be horizontal.

Learn how to find the derivative of a quadratic function, f(x) = x^2 - 2x, and calculate f'(1) with detailed step-by-step solutions. This calculus problem covers key concepts such as derivatives and the power rule. Suitable for students in Grades 10-12.

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