The Tangent routine returns the tangent to a curve at a given point.

Where the tangent is vertical, an equation form is returned.

You can also learn about tangents using the TangentTutor command.

The inverse of a function can be plotted using the InversePlot routine.  The default plot domain and range are chosen to the display reasonable portions of the function and its inverse.

You can also plot the inverse of a function using the InverseTutor command.

One reason for studying derivatives is to get qualitative information about a function.  The easiest way to sketch a function is to sample it at a number of points and connect the dots.  For example, sampling the function $\sin \left(12x\right)$ at the points x = $0\,1\,2\,3\,4$, and $5$ suggests the following approximation (shown in orange). Knowing that the sine function oscillates, you may be satisfied with this result.  The actual expression is plotted in blue.

In the following example, the global cubic behavior is very well approximated by the sampling, but the asymptote at $x\=-1$ is missed.

In other cases, some of the behavior of the expression occurs outside the sampling region. The following misses that the expression goes to $\infty$, and not $-\infty$ as the plot suggests.