Prerequisites: Definition of Derivative, Derivative Function

Goal: To conjecture the power rule.

The blue curve represents the graph of f(x)=xn, where n is a constant chosen by the vertical slider. The green line represents the tangent line at the point chosen by the horizontal slider. The red curve represents the derivative function f'(x) of f(x).

  1. Using the vertical slider, adjust the function so that f(x) = x2. 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 x between about -2.5 and 2.5.
    1. Using the grid in the background, estimate the slope of the (green) tangent line at the point you chose.
    2. 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.)
    3. Move the point of tangency twice more and repeat parts a & b.
    4. What is the relationship between the slope of the tangent line and the value of the red line?
    5. The red curve is called the derivative of f(x), often written f'(x). In the case of (blue curve) f(x) = x2, the derivative (red curve) f'(x) appears to have a linear graph. Assuming it is a line, find its equation.
    6. 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?
  2. Using the vertical slider, adjust the function so that f(x) = x3. 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 x between about -1.4 and 1.4.
    1. Using the grid in the background, estimate the slope of the (green) tangent line at the point you chose.
    2. 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.)
    3. Move the point of tangency twice more and repeat parts a & b.
    4. What is the relationship between the slope of the tangent line and the value of the red line?
    5. The red curve is called the derivative of f(x), often written f'(x). In the case of (blue curve) f(x) = x3, the derivative (red curve) f'(x) appears to have a parabolic graph. Assuming it is a parabola, find its equation.
    6. Where the tangent line is horizontal, what do you notice about the derivative?
  3. Using the vertical slider, adjust the function so that f(x) = x4. 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 x between about -1.2 and 1.2.
    1. Using the grid in the background, estimate the slope of the (green) tangent line at the point you chose.
    2. 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.)
    3. Move the point of tangency twice more and repeat parts a & b.
    4. What is the relationship between the slope of the tangent line and the value of the red line?
    5. The red curve is called the derivative of f(x), often written f'(x). In the case of (blue curve) f(x) = x4, the derivative (red curve) f'(x) appears to have a cubic graph. Assuming it is of the form y=k·x3, find its equation.
    6. Where the tangent line is horizontal, what do you notice about the derivative?
  4. Fill in the table:
    equation of f(x) apparent f'(x)
    x2 _____?
    x3 _____?
    x4 _____?
  5. Conjecture the derivative of f(x)=xn.
  6. Test your conjecture by checking the other functions that the applet displays. In particular, does your conjecture work for x-1 (=1/x), x0 (=1), and x1 (=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?
  7. Hopefully, you now have a working conjecture of the derivative of functions of the form f(x)=xn. Might there be any reasonable restrictions on 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?