Higher-Order Derivatives

If is a differentiable function, then its derivative is also a function, so may have a derivative of its own, denoted by This function is called the second derivative of Moreover, the second derivative may be differentiable, and etc.

Definition (Higher-Order Derivatives) Suppose and are differentiable functions, then the second derivative of is defined as and is denoted by Further, the third derivative is defined as and is denoted by ; and the fourth derivative is defined as and is denoted by , provided these functions exist. In general, if is differentiable, then is the derivative of

In Leibniz notation the first, second the third derivatives are

and

The derivative is denoted by and in Leibniz notation:

Example (Higher-Order Derivatives) Find the first, second, and third derivatives of

    Solution. We could use the product rule but since we want higher order derivatives it will be quicker to expand first. We find,

Thus,

Example (Higher-Order Derivatives) Find the first, second, and third derivatives of

    Solution. To find the first derivative we use the quotient rule with and   Since, and we have,


Similarly, we use the quotient rule to find the second derivative,

Similarly, we use the quotient rule to find the third derivative,