Higher Order Derivatives:

The derivative ƒ' (x) of a function y = ƒ(x) is known as the first derivative of ƒ(x) at x. consider, the function ƒ' (x) is itself differentiable at x. Then the function d/dxƒ' (x) is known as the second derivative of ƒ(x) at x. It is shown as ƒ"(x) or ƒ(2) (x) or  or y². In common, the nth derivative of ƒ(x), denoted ƒ(n)(x), is described as

The order sequence of the nth derivative of y = ƒ(x) are   or y_n or Dⁿy.

The process of searching out the nth derivative (n ≥ 2) of a function is known as successive differentiation.

Example: If  y = tan^-1 x, prove that