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What’s the relationship between the point and the definition of the derivative

The image shows a graph of a function \( f(x) \) with key points labeled \( f(a) \), \( a \), and \( a + h \), and an interactive slider controlling \( h \). This setup directly illustrates the geometric interpretation of the derivative of a function at a point.

### Relationship to the Definition of Derivative:
1. **Secant Line Slope**: The two points \( (a, f(a)) \) and \( (a + h, f(a + h)) \) are shown on the curve. The line through these points represents the secant line. The slope of this secant line is given by the difference quotient:
   \[
   \text{slope of secant} = \frac{f(a+h) - f(a)}{h}
   \]
   This slope represents the average rate of change of the function over the interval \( [a, a + h] \).

2. **Derivative Definition**: The derivative of the function at the point \( a \), denoted \( f'(a) \), is defined as the limit of the slope of the secant line as \( h \) approaches zero:
   \[
   f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
   \]
   As \( h \) gets smaller, the secant line approaches the tangent line at \( a \), and its slope approaches the derivative at \( a \).

3. **Visualization of \( h \)**: The value of \( h \) can be adjusted, as shown by the slider labeled with 0.7, which controls the distance between \( a \) and \( a + h \). Decreasing \( h \) makes the secant line approach the tangent line at \( a \), and its slope approaches \( f'(a) \).

In summary, this diagram illustrates how the slope of the secant line between two points on the curve approaches the slope of the tangent line at \( a \) as \( h \) decreases, which is the definition of the derivative.

Would you like more details or have any questions?

### Follow-up Questions:
1. How does the value of \( h \) affect the slope of the secant line?
2. What happens to the secant line when \( h = 0 \)?
3. How is the derivative interpreted when the function has a local maximum or minimum at \( a \)?
4. Can the derivative be negative based on this diagram? What does that mean geometrically?
5. How would you estimate the value of the derivative at \( a \) from this graph?

### Tip:
When learning derivatives, focus on understanding the difference between the secant line (average rate of change) and the tangent line (instantaneous rate of change).

What’s the relationship between the point and the definition of the derivative?

Explore the relationship between the secant line and the definition of the derivative in this interactive setup. Learn how the slope of the secant line approaches the tangent line as h approaches 0, a core concept in calculus. Suitable for students in Grades 10-12 studying derivatives and limits.

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