Differentiation Interactive Applet - products and quotients
You can use this interactive applet to explore some of the differentiation examples found elsewhere in this chapter.
Most of the time when we are differentiating these complicated expressions, we don't know what the original function looks like, or what we've actually found. This applet lets you see and explore some of the differentiation examples we've done.
Things to Do
In this applet, there are pre-defined examples in the pull-down menu at the top. The examples are taken from this chapter.
In the left pane you will see the graph of the function of interest, and a triangle with base 1 unit, indicating the slope of the tangent. In the right pane is the graph of the first derivative (the dotted curve).
- Select any of the examples in the pull-down menu.
- Use the slider at the bottom to change the x-value. You can drag the slider left or right (keep the cursor within the light gray region) or you can animate the points by holding down the "−" or "+" buttons either side of the slider.
See below the graphs for some notes on each of the examples.
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These examples come from Derivatives of Products and Quotients section.
The first example is a hyperbola, `y=1/x`. It has a discontinuity at `x=0`. (This means there is a "gap" in the curve at that point.) The slope of the curve is negative for all values of `x` (except `0`).
The derivative is `dy/dx=-1/x^2`, which you can see in the right hand graph. The derivative curve is completely below the x-axis.
The second example is a quotient, or one expression divided by another: `y=(2x^3) / (4-x)`.
This has a discontinuity (gap) when `x=4`. Either side of that discontinuity, the y-values go off to `+-` infinity.
The derivative is `dy/dx=(-4x^3+24x^2)/((x-4)^2)`, which also has a discontinuity at `x=4`.
Next we have `y=(4x^2) / (x^3+3)` which has a discontinuity where `x=root3(-3)~~-1.442`.
The derivative is `dy/dx=(-4x(x^3-6))/((x^3+3)^2)`, which of course will also have a discontinuity at `x~~1.442`.
Next up is `y=(x^2+3)^5` which is actually a polynomial (with power 10), and has no discontinuity.
The derivative is `dy/dx=10x(x^2+3)^4`, which is also a polynomial (with power 9).
This one involves a square root: `y=sqrt(4x^2-x)` which is actually undefined for `0<=x<=1/4`. You'll see the graph say "undefined" in this region.
The derivative is `dy/dx= (8x-1)/(2sqrt(x(4x-1)))`, which also has no meaning between `x=0` and `x=1/4`.
The next one is a polynomial: `y=(2x^3-1)^4`, with derivative `dy/dx= 24x^2(2x^3-1)^3`.
The last one (with interesting "steps") is a quotient: `y=(x^2(3x+1))/(x^4+2)`. It has derivative `dy/dx= -(x(3x^5+2x^4-18x-4))/(x^4+2)^2`.
Credits: Interactive based on a Java applet by David Eck and team from the Hobart and William Smith Colleges.