Given a function (specified in the box labeled "$f(x)=$"), this demo plots $y=f(x)$ (in blue) and a dot at $(x,y)=(a,f(a))$ (according to the value of $a$ specified by the slider). Then the derivative $f'(x)$ is computed and plotted in green with a corresponding dot at $x=a$. If the "Show second derivative?" box is checked, the second derivative is computed and plotted in orange (with a dot at $x=a$) as well.
Derivative information is displayed below the plot. Recall that if $f'(x)<0$ on an interval surrounding $x=a$, then $f(x)$ is decreasing around $x=a$. Likewise, if positive, $f(x)$ is increasing. If the derivative is zero or does not exist, we have a critical point. Keep in mind that we are rounding computations to 5 digits. This approximate calculation makes it very difficult to get Sage to hit zero exactly (it's hard to precisely hit the critical points).
If the "Show second derivative?" box is checked, the second derivative information is shown as well. Including an indication of concavity. Recall that if $f''(x)<0$ on an interval around $x=a$, then $f(x)$ is concave down around $x=a$. Likewise, if positive, $f(x)$ is concave up. If the second derivative is zero or does not exist, it could be that concavity is switching at $x=a$. If this is the case, $f(x)$ has an inflection point at $x=a$.
Several built in demos (with preset function, $a$ value, and plot domain) are provided in a drop down list. The limits of the slider bar are set using the plot domain. If you change a box entry, simply click outside the box to have sage recompute the plots and displayed values.
The computation above is powered by SageMath. The Sage code is embedded in this webpage's html file. To view the code instruct your browser to show you this page's source. For example, in Chrome, right-click and choose "View page source".