Find f^(9)(0) f(x)=x cosx^2

Name: Problem 10, Chapter 12: Calculus: One and Several Variables
Uploaded: 2021-12-04T19:40:01-08:00
Duration: 1 min 48 s
Channel: Nick Johnson
Description: Find f^(9)(0) f(x)=x cosx^2

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Key Concepts

Taylor Series Expansion

A Taylor series expresses a function as an infinite sum of terms calculated from the derivatives of the function at a specific point. In the context of evaluating a derivative at zero, the Maclaurin series—a special case of the Taylor series centered at 0—is used to represent the function as a power series, making it easier to identify the coefficients related to the derivatives.

Power Series Coefficient Extraction for Derivatives

Once a function is written as a power series, the coefficient of the term x^n is directly related to the nth derivative of the function at 0. Specifically, the nth derivative at 0 is given by n! times the coefficient of x^n in the series. This technique simplifies the computation of high-order derivatives.

Series Expansion of Composite Functions

For composite functions, such as when a function involves a composition like x and cos(x^2), the series expansion is obtained by substituting one function into the power series of another. This method allows one to effectively handle the inner function and the outer function separately before combining them to form the final series representation.

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