Derivatives Part 1

For the SL syllabus derivatives can be found using graphical reasoning. Which also provides a nice link to just what in the world a derivative really is…

Lets start with the example of the function $f(x)=x^2$. In the diagram below several points have been chosen. Tangent lines have been drawn at each point with the slope of each tangent line noted. At each x value the slope of the tangent line is different. Those values are shown below:

 x value Slope -2.5 -5 -2 -4 -1.5 -3 -1 -2 -0.5 -1 0 0 0.5 1 1 2 1.5 3 2 4 2.5 5

This new data (x versus slope) can be plotted on a new graph. All of the points clearly fall along a line with a slope of 2 and a y-intercept of 0. Thus the equation of this new line can be given by:

(1)
\begin{equation} y=2x \end{equation}

Now hold on a bit… Remembering that the derivative function $f'(x)$ is the function that tells us the slope of a tangent line at any point on a given function $f(x)$. So that means the derivative of the function $f(x)=x^2$ must be:

(2)
\begin{equation} f'(x) = 2x \end{equation}

Or…

(3)
\begin{align} \frac{dy}{dx}=2x \end{align}

For many of the more complex functions derivatives can be found in similar ways, albeit with varying success and detail… Below in a small applet that allows you to enter any function and then trace over the function while the computer plots x vs. slope (i.e. graphs the derivative function). You can select options to show the tangent and or derivative (both graphical and functional).

Right Now the Applet Embedding is a mess. Use this link: Geogebra Applet

Haven't figure out how to embed geogebra directly… just iframing for now

Doing things graphical is good and all. For you SL folk finding derivatives graphically will work for all functions but polynomials. Polynomials you must be able to differentiate from "first principles" meaning the equation! So check out derivatives part 2