The slope of the line connecting the two points $⟨a\,b⟩$ and $⟨c\,d⟩$ is given by the formula $\frac{d-b}{c-a}$.  Given a function $f\left(x\right)$, a point $x_0$, and a value, the slope of the line between the points $⟨x_0\,f\left(x_0\right)⟩$ and $⟨x_0\+h\,f\left(x_0\+h\right)⟩$ is thus $\frac{f\left(x_0\+h\right)-f\left(x_0\right)}{x_0\+h-x_0}$, which simplifies to

For a given sequence of values of $h$,

In this case, the slope approaches the value of the derivative at the point 2.0:

This convergence to the derivative can be seen using the option $\mathrm{output}\=\mathrm{animation}$.

You can also learn about Newton's Quotient using the TangentSecantTutor command.

The option order can be used to plot higher order derivatives.  In the next example, the first 4 derivatives of  $x^4-4x\+3$ are plotted, where the 1st and 4th derivatives are colored orange and green, respectively, and the colors of the intermediate derivatives are interpolated between orange and green.  (The function is plotted in blue.)

You can also plot derivatives using the DerivativeTutor command.

Calculus1, diff