Visualizing Implicit Differentiation
Discovered Definition of Implicit Differentiation
Implicit Differentiation. In implicit differentiation, we use the same definition of the derivative, where the left-hand and right-hand derivatives (LHD and RHD) approach the slope of the tangent line.
The LHD and RHD represent the limits of the secant lines, which get closer to the tangent line as the points on the curve come infinitesimally close.
For relations like a circle, where each x-value corresponds to two y-values, there are two possible tangent slopes for each x. This means we have two sets of secant lines approaching the two tangent lines, one for each y-value.
So, for relations like a circle, where a single x-value gives two y-values, we get four secant lines (two for each y) and two tangent lines.
How to Use the GeoGebra Applet:
- Explore the Secant Lines:
- Use the slider to adjust the value of a, the x-coordinate of point A.
- As you change a, notice how the two points, A=(a,y1) and B=(a,y2), move along the curve (the circle in this case).
- Adjust the Δx\Delta x slider to change the horizontal distance between the points, and watch how the secant lines change in slope as Δx\Delta x becomes smaller.
- Observe the Tangent Line:
- As the secant lines converge, observe how they approach the tangent line at each point.
- For each value of a, the secant lines will approach the slope of the tangent line at the corresponding points A and B, reflecting the dual nature of the relation (two possible y-values for each x).
- Compare Secant and Tangent Lines:
- You will see two secant lines for each y-value, one for the upper y-value and one for the lower y-value.
- These secant lines approach two different tangent lines, one for each y-value, showing how the slope of the tangent line changes depending on the position on the circle.
- Open in App
Comprehension Check Questions:
- Understanding Secant Lines:
- What happens to the secant lines as the slider for Δx\Delta x gets smaller?
- How do they behave as Δx\Delta x approaches zero?
- Exploring Tangent Slopes:
- How do the secant lines relate to the tangent line? What happens to the secant lines when they approach the tangent line at a point?
- Dual Tangent Lines for Relations:
- When x=a, why are there two possible y-values? What does this mean for the number of secant and tangent lines?
- Applying Implicit Differentiation:
- How does the concept of implicit differentiation explain the two sets of secant lines approaching the tangent lines? Can you identify both tangent lines from the secant lines?
- Generalizing to Other Relations:
- Can this concept be applied to other relations (like ellipses or other curves) where each x-value corresponds to multiple y-values? How would the number of secant and tangent lines change?