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Second Derivative

Since the derivative of a function is another function, we can take the derivative of a derivative, called the second derivative. If y = f (x), then the second derivative is written as either f '' (x) with a double prime after the f, or as d2y/dx2 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).



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See About the calculus applets for operating instructions.

1. A cubic

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 negative.

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 concave up) and when the second derivative is negative, the cubic curve is concave down (i.e., looks like concave down).

2. A Sine curve

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 curve?

3. An exponential

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?

4. A hyperbola

Select the fourth example, the hyperbola. Can you relate the concavity of the hyperbola on the left to the second derivative graph on the right?

Explore

You can also type your own function into the "f(x)=" box to explore other derivatives and second derivatives.

Other differentiation topics

Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.