Second Derivative: What Does it Represent? - James

[jamesd2008]

If the first derivative of a function represents the gradient of the tangent line...

What does the second derivative represent?

Thanks in advance
James

 

[AdrianZ]

the second derivative of some function f: R -> R represents concavity. positive values for the second derivative mean that the graph curves upward, negative values mean the graph curves downward. in physics the second derivative of position function with respect to time gives the acceleration.

 

[granpa]

rate of change of the rate of change.
like acceleration

 

[Tac-Tics]

It's the gradient of the tangent line of the derivative's graph.

 

[HallsofIvy]

I'm a little bit reluctant to say that the "first derivative of a function represents the gradient of the tangent line" or that the "second derivative of a function represents the curvature".

Certainly we can think of them that way, especially if we are focusing on graphs of functions. But functions have many different applications, we can think of functions in many different ways, and each gives a different interpretation of first and second derivative.