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 nth derivative is a formula for all successive derivatives of a function.
Examples: Finding The nth Derivative
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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) = xn
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) = nxn – 1
- f′′(x) = n(n – 1)xn – 2
- f′′′(x) = n(n – 1) (n – 2)xn – 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)xn – 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/x2
- f′′(x) = 1 ∙ 2/x3
- f′′′(x) = 1 ∙ 2 ∙ 3/x4
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!/xn + 1
Orloff, J. Differentiation.