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).