**Prerequisites:** Definition of Derivative, Derivative Function

**Goal:** To conjecture the power rule.

The blue curve represents the graph of *f(x)=x ^{n}*, where

- Using the vertical slider, adjust the function so that
*f(x) = x*. Using the lower slider, move the point of tangency along the graph from left to right. Stop anywhere and inspect both the green tangent line and the red curve; it will be helpful to chose a value of^{2}*x*between about*-2.5*and*2.5*.- Using the grid in the background, estimate the slope of the (green) tangent line at the point you chose.
- On the red curve, what is the
*y*-coordinate of the rightmost point? (Note: the rightmost point has the same*x*-coordinate as the point of tangency in part a.) - Move the point of tangency twice more and repeat parts a & b.
- What is the relationship between the slope of the tangent line and the value of the red line?
- The red curve is called the derivative of
*f(x)*, often written*f'(x)*. In the case of (blue curve)*f(x) = x*, the derivative (red curve)^{2}*f'(x)*appears to have a linear graph. Assuming it is a line, find its equation. - Where the tangent line is horizontal, what do you notice about the derivative? In other words, when
*f'(x) = 0*, what is*f(x)*doing?

- Using the vertical slider, adjust the function so that
*f(x) = x*. Using the lower slider, move the point of tangency along the graph from left to right. Stop anywhere and inspect both the green tangent line and the red curve; it will be helpful to chose a value of^{3}*x*between about*-1.4*and*1.4*.- Using the grid in the background, estimate the slope of the (green) tangent line at the point you chose.
- On the red curve, what is the
*y*-coordinate of the rightmost point? (Note: the rightmost point has the same*x*-coordinate as the point of tangency in part a.) - Move the point of tangency twice more and repeat parts a & b.
- What is the relationship between the slope of the tangent line and the value of the red line?
- The red curve is called the derivative of
*f(x)*, often written*f'(x)*. In the case of (blue curve)*f(x) = x*, the derivative (red curve)^{3}*f'(x)*appears to have a parabolic graph. Assuming it is a parabola, find its equation. - Where the tangent line is horizontal, what do you notice about the derivative?

- Using the vertical slider, adjust the function so that
*f(x) = x*. Using the lower slider, move the point of tangency along the graph from left to right. Stop anywhere and inspect both the green tangent line and the red curve; it will be helpful to chose a value of^{4}*x*between about*-1.2*and*1.2*.- Using the grid in the background, estimate the slope of the (green) tangent line at the point you chose.
- On the red curve, what is the
*y*-coordinate of the rightmost point? (Note: the rightmost point has the same*x*-coordinate as the point of tangency in part a.) - Move the point of tangency twice more and repeat parts a & b.
- What is the relationship between the slope of the tangent line and the value of the red line?
- The red curve is called the derivative of
*f(x)*, often written*f'(x)*. In the case of (blue curve)*f(x) = x*, the derivative (red curve)^{4}*f'(x)*appears to have a cubic graph. Assuming it is of the form*y=k·x*, find its equation.^{3} - Where the tangent line is horizontal, what do you notice about the derivative?

- Fill in the table:
*equation of f(x)**apparent f'(x)**x*^{2}_____? *x*^{3}_____? *x*^{4}_____? - Conjecture the derivative of
*f(x)=x*.^{n} - Test your conjecture by checking the other functions that the applet displays. In particular, does your conjecture work for
*x*,^{-1}(=1/x)*x*, and^{0}(=1)*x*? (The values -1, 0, and 1 often create mischief in mathematics.) If not, can your conjecture be adjusted so that the new data works?^{1}(=x) - Hopefully, you now have a working conjecture of the derivative of functions of the form
*f(x)=x*. Might there be any reasonable restrictions on^{n}*n*? For example, have we looked at any non-integer values such as*n=½?*or non-rational values such as*n=√2*or*n=π*?