Higher derivatives : d/dx notation and meaning

[atrus_ovis]

I understand that, having a function f(x), it's derivative function is the rate of change of f.
That df/dx means how much f changes, given an infinitesimal change in x, denoted as dx.

In second derivatives,how is d²f / dx²explained ?

Help me on the intuition please.

 

[apanna1236]

the second derivative of the function can be viewed intuitively as curvature ( how concave or convex the curve is)... if you consider the second derivative..it is the rate of change of velocity along a curve...they are very useful in obtaining the maxima and minima of a curve

 

[lurflurf]

In standard analysis dx is not infinitesimal. The second derivative is the derivative of the derivative. One can formulate a second differential, but usually it is not coordinate independent unless we require d^2=0 which is useless for the present purpose.