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What is Maclaurin's formula for the derivative of the function [tex]f(x) = x^5[/tex]?

A) [tex]f'(x) = x^5[/tex]
B) [tex]f'(x) = 5x^4[/tex]
C) [tex]f'(x) = 20x^3[/tex]
D) [tex]f'(x) = 60x^2[/tex]

Answer :

Final answer:

The question pertains to Maclaurin's series. The Maclaurin series for a function of the form f(x) = x^n is found by taking the nth derivative of the function and evaluating it at zero, then multiplying that result by the nth term in the series. None of the options in the question, however, accurately represent Maclaurin's series of x^5.

Explanation:

The student's question relates to Maclaurin’s series, which is a method to approximate a function using a series of polynomial terms. The Maclaurin series for a function is a special case of the Taylor series, centered at zero. For a function of the form f(x) = x^n, the nth derivative evaluated at zero gives a coefficient for the nth term in the series.

In the case of f(x) = x^5, the derivative, which we denote as f'(x), would be 5x^4. From there, making use of the Maclaurin's formula, we can define the terms of the series. However, strictly going by the options provided in the question, none of them represent the Maclaurin's series of x^5.

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