Differential Calculus

If dy/dx is used to find the gradient of a curve and d2y/dx2 is used to find whether a turning point is maximum or minimum, what are d3y/dx3, d4y/dx4, and so on used for?

xy2k wrote:

If dy/dx is used to find the gradient of a curve and d2y/dx2 is used to find whether a turning point is maximum or minimum, what are d3y/dx3, d4y/dx4, and so on used for?

Patiodog gave a good answer, I just wanted to explain it in a little more detail, phrase it a little differently.

You know the derivative tells you the slope, the rate of change of a function. Think about it. The second derivative gives you not only minimum and maximum points, it gives you the complete rate of change of the first derivative. The third derivative gives you the minimum and maximum points and the rate of change of the second derivative... get it?

They are used for whatever you want to use them for. Maybe in addition to knowing when the speed is at a maximum or minimum, you want to know when acceleration is at a maximum or minimum. Or when the change in acceleration is maximum or minimum.

Your question is kind of like "OK, now I know how to multiply two numbers (2x2), what good is it to multiply three or four numbers (2x2x2, 2x2x2x2, etc)??" Hmm... instead of measuring surfaces, you can now measure volumes and flows, and calculate progressive phenomena. Math does not have a meaning in itself, it is simply a tool. Be patient, and you might find uses for your new knowledge...