It is known that the first order derivative is the measure of rate of change.
This statement may be carried forward to the higher order derivatives.
But how do we visualize the higher order derivatives?
I mean suppose in a dynamics the third derivative of position arises,then how do I visualize this quantity?
In mathematics we do solve higher order differential equations.
What is the physical idea and relevance behind them?
try this link.
http://tinyurl.com/jkdvlbo
Also regarding visualizing isn't it the same as lower order derivatives? To visualize acceleration one plots velocity versus time and to view velocity it is distance versus time.
An elegant physical intuition is illustrated here: http://physics.stackexchange.com/questions/45517/physical-intuition-for-higher-order-derivatives
Sir, I mean what is the physical significance of derivative of acceleration?
You will have to see it just like accelerations is a measure of rate of change of velocity. For an extremely dynamics and sensitive system higher orders will come into play. Some examples are: http://math.stackexchange.com/questions/14841/what-is-the-meaning-of-the-third-derivative-of-a-function-at-a-point
And for your information, derivative of acceleration is called "Jerk". https://en.wikipedia.org/wiki/Jerk_(physics)
The case of the first derivative is clear. For the second one, recall that a twice differentiable function f on a convex open subset A of R^n is convex if and only if the second derivative f ' ' is positive semi - definite on A. For such functions we have f(x) -f(y)>= f ' (y)(x-y) for all x,y in A. Then we proceed by induction: f ' ' ' is positive semi - definite if and only if f ' is convex, and so on. On the other hand, If f ' ' is positive definite, then f is strictly convex. If for such a function the first derivative vanishes at a point a, then a is the unique strict minimum point of f in A. The geometric meaning of a convex (respectively strictly convex) function is well known, so that we have a geometric meaning for all derivatives.
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