Interactive Graph showing Differentiation of a Polynomial Function

In the following interactive you can explore how the slope of a curve changes as the variable `x` changes.

Things to do

In this applet, there are pre-defined examples in the pull-down menu at the top. The examples are taken from 5. Derivatives of Polynomials.

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).

  1. 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.
  2. Select another of the other 2 examples in the pull-down menu.

The height of the right triangle indicates the slope. It has a base of 1 unit.

Choose function:

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Slope

dy/dx

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Background

Here are the derivatives of the 3 functions given above:

1. Quadratic (parabola), `y=x^2-10x-1`.

Derivative: `dy/dx=2x-10`

2. Cubic, `y=0.015x^3-0.25x^2+0.49x+0.47`.

Derivative: `dy/dx=0.045x^2-0.5x+0.49`

3. Quartic `y=x^4-1.5x^3-6x^2+3.5x+3`.

Derivative: `dy/dx= 4x^3-4.5x^2-12x+3.5`

See how to find these derivatives in the Derivatives of Polynomials section.