Using the Dynamic Tangent tool to Visualize the Derivative as a Function

Use this activity to visually explore the concept of the derivative of a function. Students can visualize derivative functions before they learn the rules of differentiation.

Plot the quadratic function y1(x) = 4(x-2)(x+3) on -5x5 and -125y125. If autoscaling is checked, click it to uncheck it.
Click the Tangent tool icon to select it.
Select Dynamic from the Tangent tool's pop-up menu.
Click the Go button in the Graph window.
Watch the tangent line move along the plotted curve and the slope of the tangent line being plotted to draw the graph of the derivative of the function.

Notice that the derivative of a quadratic function is a linear function.

Note:

To change the speed of the moving tangent, select Dynamic Plotting Speed... from the Options menu.
To change the length and thickness of the moving tangent, use the Tangent tool's pop-up menu.
To move the tangent one step at a time, click the Step button.

Plot the cubic function y2(x) = 2x(x-3.5)(x+4) on -5x5 and -125y125.
Click the Go button in the Graph window.

Notice that the derivative of a cubic polynomial is a quadratic function.

Plot the fourth degree polynomial y3(x) = (x-1)(x-4)(x+4.5)(x+2)/3 on -5x5 and -125y125.
Click the Go button in the Graph window.

Notice that the derivative of a fourth degree polynomial is a third degree polynomial.

Plot the sine function y4(x) = sin(x) on -5x5 and -2y2. Note the change in the plotting range.

If the Tangent tool's icon gets deselected, simply click the icon to select it.

Click the Go button in the Graph window.

Can you guess what the derivative function is?

Enter your guess into the Work window and overlay its plot.
Does the graph of your guess and the plotted derivative match?

When is a function differentiable?

Plot the absolute value function y5(x) = abs(x) on -5x5 and -5y5.
Click the Go button in the Graph window.

Describe the derivative. What happens to the derivative as x approaches 0? Is abs(x) differentiable at x = 0? Why or why not?

Plot the function y6(x) = abs(x^2-9) on -5x5 and -20y20.
Note that you may need to activate this function by clicking to the left of the letter R. This will place a checkmark there and the function becomes plottable.
Click the Go button in the Graph window.

Describe the derivative. What happens to the derivative as x approaches -3 and 3? Is the function differentiable at x = -3 and x = 3? Why or why not?

Plot the cube root function y7(x) = rad(3, x-1) on -1x3 and -4y4.
Click the Go button in the Graph window.

Describe the derivative. What happens to the derivative as x approaches 1? Is the function differentiable at x = 1? Why or why not?

Plot the function y8(x) = -x rad(5,(x/2-1)^2)+1 on -1x3 and -4y4.
Click the Go button in the Graph window.

Describe the derivative. What happens to the derivative as x approaches 2? Is the function differentiable at x = 2? Why or why not?

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