Since the dervative of a function is another function, we can take the
derivative of a derivative, called the second derivative. Notationally if y = f (x), then the second derivative is written as
either f '' (x) or as . Higher derivatives can also be
defined. If the first derivative tells you about the rate of change of a
function, the second derivative tells you about the rate of change of the
rate of change. In physics, if s (t) is the position of a
particle at time t, then s' (t) = v (t) is the
velocity (i.e., the rate of change of the position), and v' (t)
= s'' (t) = a (t) is the acceleration (i.e., the
rate of change of the velocity).
Try the following:
- The initial example shows a cubic curve on the left, its derivative in
the middle, and the second derivative on the right. The red line is
tangent to the cubic and the slope of this curve is the value of the
derivative. Move the slide and compare the red line and the red crosshair
in the middle graph. As the slope of the red tangent becomes more
positive, the crosshair moves higher; when the slope of the red line
decreases, the crosshair drops lower.
- The green line in the middle graph is tangent to the derivative curve,
so the green crosshair in the right graph represents the value of the
slope of the green line, and hence is the second derivative (i.e., the
derivative of the derivative). Move the slider and note that when the
green line has positive slope, the second derivative is positive, but
where the green line has negative slope, the second derivative is
- As you move the slider, do you notice anything that relates the second
derivative to the cubic's graph? When the second derivative is positive,
what do you notice about the shape of the cubic in that region? When the
second derivative is negative, what is the shape of the cubic? You should
find that when the second derivative is positive, the cubic curve is
concave up (i.e., looks like ) and when the second derivative is
negative, the cubic curve is concave down (i.e., looks like ).
- Select the second example from the drop down menu, the sine curve. Move
the slider. Can you see how the sine curve, the derivative curve, and the
second derivative curve are related? As the red tangent line moves, does
the red crosshair's height represent the slope? As the green tangent line
moves, does the green crosshair's height represent its slope? Does the
graph of the second derivative tell you the concavity of the sine
- Select the third example, the exponential function. Move the slider.
Does it make sense that the second derivative is always positive? Why?
What is it about the shape of the original function that tells you the
second derivative will always be positive?
- Select the fourth example, the hyperbola. Can you relate the concavity
of the hyperbola on the left to the second derivative graph on the
- You can also type your own function into the "f(x)=" box to explore
other derivatives and second derivatives.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.