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. [br]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.[br][br]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.[br]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.

[br][list=1][*][b]Explore the Secant Lines:[/b][br][list][*]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.[/*][/list][/*][*][b]Observe the Tangent Line:[/b][br][list][*]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).[/*][/list][/*][*][b]Compare Secant and Tangent Lines:[/b][br][/*][list][*]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.[/*][/list][*][b]Open in App[br][/b][/*][/list][b] Calculate the slope of the tangent lines using the slope functionality in GeoGebra.[/b][br] Compare these slope values to the components e and l.[br] Kindly compare these findings to the Implicit Differentiation function at the element [br] b(x,y)=ImplicitDerivative(x2+y2−25).

[br][list=1][*][b]Understanding Secant Lines:[/b][br][list][*]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?[/*][/list][/*][*][b]Exploring Tangent Slopes:[/b][br][list][*]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?[/*][/list][/*][*][b]Dual Tangent Lines for Relations:[/b][br][list][*]When x=a, why are there two possible y-values? What does this mean for the number of secant and tangent lines?[/*][/list][/*][*][b]Applying Implicit Differentiation:[/b][br][list][*]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?[/*][/list][/*][*][b]Generalizing to Other Relations:[/b][br][list][*]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?[/*][/list][/*][/list]