A derivative is the steep of a tangent heat at a point. Here, the derivatives at points A and also B are zero.

You are watching: Write a formula for the nth derivative of the functions.

The nth derivative ad to any one the a number of greater level derivatives the a function.

When you take the derivative of role one time, you obtain the an initial derivative. Differentiating the new function one more time gives you the second derivative. Likewise, a third, fourth or fifth application of the rule of differentiation provides us the third derivative, 4th derivative and fifth derivative, respectively. The nth derivative is a formula for all succeeding derivatives of a function.

Examples: recognize The nth Derivative

Finding the nth derivative method to take a few derivatives (1st, 2nd, 3rd…) and also look because that a pattern. If one exists, climate you have a formula for the nth derivative.In bespeak to find the nth derivative, discover the first couple of derivatives to identify the pattern. Use the usual rules the differentiation come a function. Find each succeeding derivative to come at the nth.

Example 1: uncover the nth derivative that f(x) = xn

Since this function has exponents, use the Power dominance to uncover the first few derivatives. When you calculate the first 3-4 derivatives that the function, friend should gain a feeling of the overall pattern.

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The first three derivatives the this function are:

n(n-1) (n-2)…(n – n + 1)

You have the right to write the formula because that the nth derivative as:

f(n) = n(n – 1) (n – 2)…(n – n + 1)xn – n

The sample of successive multiplication is referred to as a factorial and is created as n!.

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

Find the an initial three derivatives the the duty and climate solve:

f′(x) = -1/x2f′′(x) = 1 ∙ 2/x3f′′′(x) = 1 ∙ 2 ∙ 3/x4

The pattern arising involves adding an additional consecutive number come the numerator and also another exponent come the denominator. We have the right to write the nth derivative as:

f(n) = (-1)n ∙ n!/xn + 1


Orloff, J. Differentiation.

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