When derivative of any function itself differentiable, with the further derivative . Any derivative beyond the first derivative can be referred to as a higher order derivative.
If y = f(x) then , the derivative of y with respect to x, is itself, in general , a function of x and can be differentiated again. To fix up the idea, we shall call w.r.t. x as the first order derivative of y with respect to x and
the derivative of w.r.t. as the second order derivative of y w.r.t. x and will be denoted by .
Similarly the derivative of w.r.t. x will be termed as the third order derivative of y w.r.t. x and will be denoted by
and so on. the nth order derivative of y w.r.t. x will be denoted by
If y=f(x), then the other alternative notations for
The values of these derivatives at x=a are denoted by
We have . On differentiating w.r.t. x , we get
Again differentiating w.r.t. x we get