The

**nth derivative**refers to any one of a number of higher order derivatives of a function.

When you take the derivative of function one time, you get the first derivative. Differentiating the new function another time gives you the second derivative. Likewise, a third, fourth or fifth application of the rules of differentiation gives us the third derivative, fourth derivative and fifth derivative, respectively. The *n*th derivative is a formula for all successive derivatives of a function.

## Examples: Finding The nth Derivative

Watch the video for three examples:

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**Finding the nth derivative** means to take a few derivatives (1st, 2nd, 3rd…) and look for a pattern. If one exists, then you have a formula for the nth derivative. To find the nth derivative, find the first few derivatives to identify the pattern. Apply the usual rules of differentiation to a function, then find each successive derivative to arrive at the *nth*.

**Example 1**: Find the nth derivative of f(x) = x^{n}

Since this function has exponents, use the Power Rule to find the first few derivatives. Once you calculate the first 3-4 derivatives of the function, you should get a sense of the overall pattern.

The first three derivatives of this function are:

- f′(x) = nx
^{n – 1} - f′′(x) = n(n – 1)x
^{n – 2} - f′′′(x) = n(n – 1) (n – 2)x
^{n – 3}

**A pattern is emerging:** Each successive derivative adds another layer of n minus (n -), which can be written as:

n(n-1) (n-2)…(n – n + 1)

You can write the formula for the nth derivative as:

f(n) = n(n – 1) (n – 2)…(n – n + 1)x^{n – n}

The pattern of successive multiplication is called a **factorial** and is written as n!.

**Example 2**: Find the nth derivative of f(x) = 1/x

Find the first three derivatives of the function and then solve:

- f′(x) = -1/x
^{2} - f′′(x) = 1 ∙ 2/x
^{3} - f′′′(x) = 1 ∙ 2 ∙ 3/x
^{4}

The pattern emerging involves adding an additional consecutive number to the numerator and another exponent to the denominator. We can write the nth derivative as:

f(n) = (-1)^{n} ∙ n!/x^{n + 1}

## References

Orloff, J. Differentiation.

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