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Successive Differentiation

Let y = f (x) be a differentiable function of x. Then its derivative f ‘(x), is called the first derivative of f. The fist derivative, in general is itself a function of x which can further be differentiated with respect to x. The resulting derivative, if it exists, is called the second derivative of f. It is denoted by f ‘’(x) (read “f double prime of x”) . Similarly, if the second derivative is differentiable, its derivative is called the third derivative, written f ‘’’(x). Continuing in this way, we get higher-order derivatives. The process of finding higher-order derivatives of a function is called successive differentiation.

Some other notations for higher-order derivatives are:

First derivative        :         y ', y1, dy/dx, d/dx [f (x)]

Second derivative   :          y ' ', y2' d2y/dx2' d2/dx2 [f (x)]

Third derivative       :          y ' ' ', y3, d3y/dx3, d3/dx3 [f (x)].

nth derivative          :          y(n) , yn, dny/dxn, dn/dxn [f (x)].

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