Real Function Derivative Visualizer: Polar Domain Coloring Plot
This plot generates three images:
On the top, \(f(x)\), with a movable point \(a\).
The center shows \(f(x)-f(a)\). This shifts \(f(x)\) so that there is a root at \(a\).
In this center plot, if \(f\) is differentiable, we can see the the function appears to look like a linear function around \(a\):
It has a single band from cyan to red around the root at \(a\)
On the bottom is a plot of \(f'(x)\), with the value of \(f'(a)\) a marked on the graph.
When \(f'(a)\) is negative, notice that the direction of colors around \(f(a)\) in the center graph reverse.
If you look very carefully at the 'radius' of the color swirl around \(a\) in the center plot, you may also notice that a dilation by \(|f'(a)|\) is also occurring.
However, the transformation appears to be 'reversed,' with a dilation of 2 causing the color radius to shrink by 1/2. This happens because because this is a domain color plot, rather than a range color plot. In a range color plot each point \(z\) stays the same color, but is transformed and moved to a new location \(f(z)\), so this shows the transformation directly. In a domain color plot, each point \(f(z)\) is left in its original location \(z\) and simply given a new color. Because the points are re-colored, rather than moved, points that are the same color don't correspond, and the transformation appears to be in the opposite direction.
Therefore, we can interpret the value of \(f'(a)\) as a scaling and rotation from \(1\), indicated by the angle, with a rotation of 180° when \(f'(a)<0\)
When you drag point \(a\) on the left graph, the remaining graphs will update to show the derivative at the new \(a\).
Please be patient, this update may take some time.
This plotter has some complex output support, and you can see that complex valued derivatives will create rotations other than 180°. For example, try \(\sqrt{x}\) when \(x\)<0
Enter your function here:
\(f(x)= \) →
\(|x| \in\) [0,]
\(|x|\) scale